Evolution is one of the greatest theories in all of science. It sets out to explain life: specifically, how the first simple life gave rise to all the huge diversity we see today, from bacteria to oak trees to blue whales.

For scientists, evolution is a fact. We know that life evolved with the same certainty that we know the Earth is roughly spherical, that gravity keeps us on it, and that wasps at a picnic are annoying.

Not that you would know that from the media in some countries, where evolution is ferociously argued about – put down as "just a theory" or dismissed as a flat-out lie.

Why are biologists so certain about this? What is the evidence? The short answer is that there is so much it's hard to know where to start. But here is a very cursory summary of the evidence that life has, indeed, evolved.

It might help to first spell out quickly what Darwin's theory of evolution actually says. Most of us have the general idea: species change over time, only the fittest survive, and somehow a monkey-like creature gave rise to human beings.

It is hard to accept that you are descended, through countless generations, from a worm

Darwin's theory of evolution says that each new organism is subtly different from its parents, and these differences can sometimes help the offspring or impede it. As organisms compete for food and mates, those with the advantageous traits produce more offspring, while those with unhelpful traits may not produce any. So within a given population, advantageous traits become common and unhelpful ones disappear.

Given enough time, these changes mount up and lead to the appearance of new species and new types of organism, one small change at a time. Step by step, worms became fish, fish came onto land and developed four legs, those four-legged animals grew hair and – eventually – some of them started walking around on two legs, called themselves "humans" and discovered evolution.

This can be hard to believe. It's one thing to realise that you are not identical to our parents: perhaps your hair is a different colour, or you are taller, or have a more cheerful nature. But it is much harder to accept that you are descended, through countless generations, from a worm.

Plenty of people certainly don't accept this. But forget all the drama for a moment. Instead, begin as Charles Darwin did: on your doorstep.

Darwin's book On the Origin of Species, first published in 1859, begins by asking the reader to look around at the familiar. Not unexplored tropical islands or faraway jungles, but the farmyard and garden. There, you can easily see that organisms pass on characteristics to their offspring, changing the nature of that organism over time.

These changes from generation to generation are called "descent with modification"

Darwin was highlighting the process of cultivation and breeding. For generations, farmers and gardeners have purposefully bred animals to be bigger or stronger, and plants to yield more crops.

Breeders work just like Darwin imagined evolution worked. Suppose you want to breed chickens that lay more eggs. First you must find those hens that lay more eggs than the others. Then you must hatch their eggs, and ensure that the resulting chicks reproduce. These chicks should also lay more eggs.

If you repeat the process with each generation, eventually you'll have hens that lay far more eggs than wild chickens do. A female jungle fowl – the closest wild relative of the domestic chicken – might lay 30 eggs in a year, whereas farm hens may well produce ten times as many.

These changes from generation to generation are called "descent with modification".

Our oldest domesticated animals are still capable of rapid improvement or modification

A young chick will in many ways be similar to its parents: it will be recognisably a chicken, and definitely not an aardvark, and it will probably be more similar to its parents than it is to other chickens. But it won't be identical.

"That's what evolution is," says Steve Jones of University College London in the UK. "It's a series of mistakes that build up."

You might think that breeding can only make a few changes, but there seems to be no end to it. "No case is on record of a variable being ceasing to be variable under cultivation," wrote Darwin. "Our oldest cultivated plants, such as wheat, still often yield new varieties: our oldest domesticated animals are still capable of rapid improvement or modification."

Breeding, Darwin argued, is essentially evolution under human supervision. It shows us that the tiny changes from generation to generation can add up. "It's inevitable," says Jones. "It's bound to happen."

Still, it's quite a step from carefully breeding chickens that lay more eggs to the natural evolution of new species. According to evolutionary theory, those chickens are ultimately descended from dinosaurs, and if you go further back, from fish.

The answer is simply that evolution takes a long time to make big changes. To see evidence of that, you have to look at older records. You have to look at fossils.

Fossils are the remains of long-dead organisms, preserved in rock. Because rocks are laid down in layers, one on top of the other, the fossil record is generally set out in date order: the oldest fossils are at the bottom.

I always think that the most convincing case for evolution is in the fossil record

Running through the fossil record makes it clear that life has changed over time.

The oldest fossils of all are the remains of single-celled organisms like bacteria, with more complicated things like animals and plants only appearing much later. Among the animal fossils, fish appear much earlier than amphibians, birds or mammals. Our closest relatives the apes are only found in the shallowest – youngest – rocks.

"I always think that the most convincing case for evolution is in the fossil record," says Jones. "It's noticeable that one page in every six in the Origin of Species is to do with the fossil record. [Darwin] knew that that was an irrefutable case that evolution had taken place."

How do we really know that one species evolved into another?

By carefully studying fossils, scientists have been able to link many extinct species with ones that survive today, sometimes indicating that one descended from another.

For example, in 2014 researchers described the fossils of a 55-million year old carnivore called Dormaalocyon, which may be a common ancestor of all today's lions, tigers and bears. The shapes of Dormaalocyon's teeth gave it away.

Still, you may not be convinced. Those animals may all have similar teeth, but lions, tigers and Dormaalocyons are still distinct species. How do we really know that one species evolved into another?

The fossil record is only so much help here, because it is incomplete. "If you look at most fossil records, what you actually see is one form that lasts quite a long time and then the next bunch of fossils that you've got is quite different from what you had before," says Jones.

It is also possible to observe the evolution of a new species as it happens

But as we have dug up more and more remains, a wealth of "transitional fossils" has been discovered. These "missing links" are halfway houses between familiar species.

For instance, earlier we said that chickens are ultimately descended from dinosaurs. In 2000 a team led by Xing Xu of the Chinese Academy of Sciences described a small dinosaur called Microraptor, which had feathers similar to modern birds and may have been able to fly.

It is also possible to observe the evolution of a new species as it happens.

In 2009, Peter and Rosemary Grant of Princeton University in New Jersey described how a new species of finch came into being on one of the Galápagos Islands: the same islands visited by Darwin.

This little group of birds had formed a new species

In 1981, a single medium ground finch arrived on an island called Daphne Major. He was unusually large and sang a somewhat different song to the local birds.

He managed to breed, and his offspring inherited his unusual traits. After a few generations, they were reproductively isolated: they looked different from the other birds, and sang different songs, so could only breed among themselves. This little group of birds had formed a new species: they had "speciated".

This new species is only subtly different from its forebears: their beaks are different and they sing an unusual song. But it is possible to watch far more dramatic changes as they happen.

Richard Lenski of Michigan State University is in charge of the world's longest-running evolution experiment.

It's a very direct demonstration of Darwin's idea of adaptation by natural selection

Since 1988, Lenski has been tracking 12 populations of Escherichia coli bacteria in his lab. The bacteria are left to their own devices in storage containers, with nutrients to feed on, and Lenski's team regularly freezes small samples.

The E. coli are no longer the same as they were in 1988. "In all 12 populations, the bacteria have evolved to grow much faster than did their ancestor," says Lenski. They have adapted to the specific mix of chemicals he gives them.

"It's a very direct demonstration of Darwin's idea of adaptation by natural selection. Now, 20-some years into the experiment, the typical lineage grows about 80% faster than did the ancestor."

In 2008, Lenski's team reported that the bacteria had made a huge leap forward. The mixture they live in includes a chemical called citrate, which E. coli cannot digest. But 31,500 generations into the experiment, one of the 12 populations started feeding on citrate. This would be like humans suddenly developing the ability to eat tree bark.

All living things carry genes, in the form of DNA

The citrate was always there, says Lenski, "so all of the populations have [had] the opportunity in a sense to evolve the ability to use this... But only one of the 12 populations has found their way to do this."

At this point, Lenski's habit of regularly freezing samples of the bacteria proved crucial. He was able to go back through older samples, and trace the changes that led to the E. coli eating citrate.

To do this, he had to look under the hood. He used a tool that wasn't available in Darwin's day, but which has revolutionised our understanding of evolution as a whole: genetics.

All living things carry genes, in the form of DNA.

Genes control how an organism grows and develops, and they are passed on from parent to offspring. When a mother chicken lays lots of eggs, and passes that trait onto her offspring, she does so through her genes.

All modern life has descended from a single common ancestor

Over the last century scientists have catalogued the genes from different species. It turns out that all living things store information in their DNA in the same way: they all use the same "genetic code".

What's more, organisms also share many genes. Thousands of genes found in human DNA may also be found in the DNA of other creatures, including plants and even bacteria.

These two facts imply that all modern life has descended from a single common ancestor, the "last universal ancestor", which lived billions of years ago.

By comparing how many genes organisms share, we can figure out how they are related. For instance, humans share more genes with apes like chimps and gorillas than other animals, as much as 96%. That suggests they are our closest relatives.

We have a common ancestor with chimpanzees

"Try to explain that in any other way than the fact that those relationships are based on a sequence of changes through time," says Chris Stringer of the Natural History Museum in London. "We have a common ancestor with chimpanzees, and we and they have diverged since then from that common ancestor."

We can also use genetics to track the detail of evolutionary changes.

"You can compare different types of bacteria and find the genes that they share," says Nancy Moran at the University of Texas at Austin. "Once you recognise these genes… you can look at how they have evolved in different kinds of populations."

When Lenski went back through his E. coli samples, he found that the citrate-eating bacteria had several changes to their DNA that the other bacteria didn't. These changes are called mutations.

Lenski's E. coli show us that evolution can give organisms radically new abilities

Some of them had happened long before the bacteria developed their new ability. "In and of themselves, [these mutations] did not confer the ability to grow on citrate, but set the stage for subsequent mutations that then conferred that ability," says Lenski.

This complex chain of events helps explain why only one population evolved the ability.

It also illustrates an important point about evolution. A particular evolutionary step may seem extremely unlikely, but if there are enough organisms being pushed to take it, one of them probably will – and it only takes one.

Lenski's E. coli show us that evolution can give organisms radically new abilities. But evolution doesn't always make things better. Its effects are often, to our eyes at least, rather random.

The mutations that lead to changes in an organism are very rarely for the better, says Moran. In fact, most mutations have either no impact, or a negative impact, on the way an organism functions.

Animals that live in dark caves often lose their eyes

When bacteria are confined to isolated environments, they sometimes pick up unwelcome genetic mutations that get passed on directly to every generation. Over time, this gradually hampers the species.

"It really shows the process of evolution," says Moran. "It's not all just adaptation and things getting better, there's also this big potential for things to get worse."

What's more, organisms sometimes lose abilities. For instance, animals that live in dark caves often lose their eyes.

This may seem odd. We tend to think of evolution as a process of biological betterment, of species improving and becoming less primitive. But this is not necessarily what happens.

The notion of betterment can be traced back to a scientist named Jean-Baptiste Lamarck, who was pushing the idea that organisms evolve before Darwin was. His contributions were vital.

What did he mean they wanted to improve? How would you test that?

But unlike Darwin, Lamarck thought that organisms got better at living in their environments as a deliberate reaction to those environments, as though they inherently wanted to improve.

Lamarck's theory would say that giraffes have long necks because their ancestors stretched to reach tall trees, and then passed their newly-acquired long necks on to their offspring.

"Darwin wrote about Lamarck privately and said his theory is complete nonsense, it's untestable," says Jones. "What did he mean they wanted to improve? How would you test that?"

Darwin had an alternative theory: natural selection. It offers a completely different explanation for giraffes' long necks.

Imagine an ancestor of modern giraffes, something a bit like a deer or antelope. If there were lots of tall trees where this animal lived, the animals with the longest necks would get more food, and do better than those with shorter necks.

Animals like giraffes are so striking because they appear so perfectly adapted

After a few generations, all the animals would have slightly longer necks than their ancestors did. Again, those with the longest would do best, so over many years, giraffes' necks would gradually get longer, because those with short necks tended not to have offspring.

The mutations underlying this all happened at random, and were just as likely to produce short necks as long ones. But those short-neck mutations didn't tend to last.

Animals like giraffes are so striking because they appear so perfectly adapted. They live in areas where the trees are tall and only have leaves high off the ground, so of course they have long necks to reach them.

"That kind of image is actually what confuses people, I think, because it looks so perfect, it looks designed," says Moran. But if you look closer, it is the result of a long chain of little changes. "You realise, oh, it's not designed, it's actually one odd event that might have spread and led to another odd event."

We now have all the pieces of evidence that, when put together, show that life has evolved.

Human evolution has always been a concept difficult for some to stomach

Descent with modification, which is caused by random mutations in genes, ultimately leads to gradual changes and the formation of new species – much of it driven by natural selection, which weeds out those organisms that are less suited to their environments.

Finally, let's apply all this to ourselves.

Human evolution has always been a concept difficult for some to stomach, but it's impossible to turn a blind eye to it now, says Stringer.

Homo sapiens is believed to have evolved in Africa before spreading all over the world.

People of European and Asian descent carry Neanderthal genes in their DNA

The fossil record shows a gradual change from ape-like animals walking on all fours to bipedal creatures that gradually developed bigger brains.

The first humans to leave Africa interbred with other hominin species, such as the Neanderthals. As a result, people of European and Asian descent carry Neanderthal genes in their DNA, but people of African descent don't.

This all happened thousands of years ago, but the story is not over. We are still evolving.

For instance, in the 1950s a British doctor called Anthony Allison was studying a genetic disorder called sickle-cell anaemia, which is common in some African populations. People with the disorder have misshapen red blood cells, which don't carry oxygen around the body as well as they might.

For those people, it was worth carrying the sickle-cell mutation

Allison discovered that the east African populations were divided into groups of lowland-dwelling people, who were prone to the disease, and people who lived in the highlands, who were not.

It turned out that people carrying the sickle-cell trait got an unexpected benefit. It protected them from malaria, which was only really a threat in the lowlands. For those people, it was worth carrying the sickle-cell mutation, even if their children might be anaemic.

By contrast, people living in highland areas were not at risk from malaria. That meant there was no advantage to carrying the sickle-cell trait, so its otherwise-harmful nature had meant it disappeared.

Of course, there are all sorts of questions about evolution that we still haven't answered.

Their ancestors go back in an unbroken line for over 3 billion years

Stringer offers a simple one: what was the genetic change that allowed humans to walk upright, and why was that mutation so successful? Right now we don't know, but with more fossils and better genetics, we might someday.

What we do know is that evolution is a fact of nature. It is the basis for life on Earth as we know it.

So next time you're out and about, whether it's in your garden or on a farm or just walking down a road, take a look at the animals and plants around you and think about how they all got there.

Each of the organisms you see, whether it's a tiny insect or a great big elephant, is the latest member of an ancient family. Their ancestors go back in an unbroken line for over 3 billion years, to the dawn of life itself. So do yours.