Immortality can be a curse rather than a blessing – as Tithonus learned to his cost.

This mythical Trojan prince was so handsome that he bewitched Eos, the goddess of dawn. She successfully petitioned Zeus to grant Tithonus immortality so she could be with him forever.

But Zeus interpreted her request literally. Tithonus didn't die, but he did age. He lost his good looks and his faculties, and Eos lost her interest. She eventually shut him away in a room where he babbles endlessly.

It's just a story. But as is often the case, truth turns out to be stranger than fiction. Plenty of species really are technically immortal. And unlike Tithonus, many are eternally youthful to boot.

What we're talking about here is "biological immortality", although many biologists would probably rather we didn't use the phrase.

Biologically immortal organisms do die, but they don't seem to age

"Immortal really means you don't die at all, which is stupid," says Thomas Bosch at the University of Kiel, Germany.

Paradoxical though it might seem, biologically immortal organisms are definitely mortal. They can be killed by a predator, a disease, or a catastrophic change in the environment such as an erupting volcano. But unlike humans, they rarely die simply because they get old.

To put it another way, biologically immortal organisms do die, but they don't seem to age. They're basically the exact opposite of Tithonus.

The bristlecone pine is a good example. Some of these North American trees are astonishingly old. They began growing 5000 years ago: about the time the real city of Troy was founded in what is now Turkey.

An old bristlecone pine looks old

As far as external appearances go, the years have been about as kind to the bristlecone pines as they will have been to Tithonus.

"The trees are pretty beaten up," says Howard Thomas at Aberystwyth University in the UK. "They get struck by lightning, weighed down by heavy snow fall, branches snap off."

In other words, an old bristlecone pine looks old. But look more closely and it's a different story.

A study published in 2001 compared pollen and seeds from bristlecone pines of various ages up to 4700 years, and found no significant increase in mutation rates with age. What's more, the vascular tissue functioned just as well in ancient trees as in juveniles.

The stem cells can apparently remain youthful and vigorous for millennia

Old trees are weather-beaten and gnarled, but at the cellular level they appear to be as youthful as they were when Troy was being built. Their tissues don't seem to be withered by such vast expanses of time.

No one really knows how the bristlecone pine does it. Their longevity isn't as well studied as you might expect. But Thomas thinks it probably comes down to a special property of the trees' "meristems".

These are bits of the roots and shoots that are home to populations of stem cells, which generate new growth. The stem cells can apparently remain youthful and vigorous for millennia.

"You do get mutations, things can go wrong," says Thomas. "But like a bacterial culture, the non-mutated cells appear to outperform the damaged ones."

There's another possibility, says Lieven De Veylder at Ghent University in Belgium. He thinks a key factor might be a small population of cells in plant meristems called the "quiescent centre".

These secret tricks of the meristem don't help most plants achieve immortality

Here, cells divide at a much reduced rate, and that might suppress division of the meristem stem cells too. That could be useful, because every time a cell divides it runs the risk of incorporating a dangerous mutation into its DNA. "Keeping a subpopulation of stem cells that divide only infrequently might be a way to keep a close-to-perfect 'back-up' genome," says De Veylder.

In 2013 his team identified a protein that seems to control activity in the quiescent centre of a plant called Arabidopsis. Similar proteins might help plants like the bristlecone pine avoid cellular ageing, allowing some of them to live for thousands of years.

However, these secret tricks of the meristem don't help most plants achieve immortality. That's because they live too fast.

"A wave of senescence can overrun the behaviour of the meristem, and you have an annual or biennial plant," says Thomas. In essence, the cells of plants like Arabidopsis work and divide so quickly, their organs burn out before the meristem can replenish the damaged tissue.

Ming the mollusc is the oldest verified solitary animal on record

By contrast, the biologically immortal plants live at a more measured pace. "Overlaying the meristem activity is a pattern of individual organ longevity," says Thomas.

When it comes to living fast, plants generally have nothing on animals. That may be why animals rarely manage more than a few centuries before they die. There's one exception: colonial animals like corals can survive for more than 4000 years. However, the individual coral polyps may be only a few years old.

Ming the mollusc is the oldest verified solitary animal on record. This ocean quahog was 507 years old when biologists dredged it up from the coastal waters around Iceland in 2006, and promptly killed it.

Ming died, but it might have been biologically immortal. In many animal cells, oxygen-containing molecules react with the membranes, generating small molecules that in turn damage other parts of the cell.

Not all animals carry a nice convenient record of their age around with them

But a 2012 study found that ocean quahog cells carry membranes that are unusually resistant to this sort of damage. Ming might have lived so long because its cells, like the cells of bristlecone pine, aged at a negligible rate.

Ming is the oldest animal with an age that can be verified. It's a mollusc, so biologists can count annual growth lines in its shell, much as botanists can age trees by counting rings in the trunk.

Not all animals carry a nice convenient record of their age around with them. Some of these might be even older than Ming.

Take the Hydra, a tiny soft-bodied animal related to jellyfish. Small animals generally don't live as long as large ones, but one biologist has kept individual Hydra in the lab for more than four years. That's an astonishingly long time for an animal that generally measures just 15mm.

Perhaps a few years is about all most manage before succumbing

What's more, at the end of the four-year experiment the Hydra looked as youthful as on day one. That makes Hydra another case of biological immortality.

Exactly how long individual Hydra might live is anyone's guess. Perhaps a few years is about all most manage before succumbing to threats like disease. Or perhaps Hydra can live for 10,000 years.

A few years ago Bosch and his colleagues offered an explanation for Hydra's lack of cellular ageing. Put simply, he says, it again comes down to stem cells.

Hydra carries a remarkably potent set of stem cells in its tiny body. They are so potent, they can regrow significant chunks of the Hydra's body in the event of an accident. This ability earned the Hydra its name, inspired by the mythological Hydra of Lerna, which could supposedly re-grow decapitated heads.

If you knock out the FoxO gene, you make Hydra age

The real world Hydra's regenerative powers are more than a mere party trick: they are crucial during reproduction. Hydra doesn't usually reproduce sexually, and instead grows tiny clones of itself.

It uses three distinct stem cell populations to replicate all of the various tissues that together form a fully functioning animal. Bosch and his colleagues have found that all three share one protein in common: FoxO. He thinks it's a key anti-ageing protein.

"If you knock out the FoxO gene, you make Hydra age," he says.

Exactly how FoxO prevents Hydra, and in particular its stem cells, from ageing isn't yet clear.

Even 100-year-old humans are not biologically immortal

But we do know it acts as a "hub" in the cell that integrates various molecular signals, including some from the cell's external environment. "We are now working on how these environmental signals are integrated with FoxO," says Bosch.

FoxO might actually be a universal anti-ageing mechanism throughout the animal kingdom. Humans carry a few versions, and some variants are more common in people who live beyond their 100th birthday.

But even 100-year-old humans are not biologically immortal: at least, not in the way that Hydra is.

Then again, the immortal jellyfish isn't biologically immortal in quite the way that Hydra is either. But it is immortal.

To understand why, it helps to know a little about the immortal jellyfish's complicated life cycle.

When jellyfish sperm and egg come together they form a tiny larva. But this larva doesn't simply grow into an adult jellyfish. Instead it usually plonks down on a hard surface and turns into a soft-bodied branching structure called a polyp.

It's as if a butterfly suddenly reverted back into a caterpillar

Most of the time these polyps produce tiny clones of themselves – just like Hydra, which is itself a polyp – but in some species the polyp also does something else. It produces small free-swimming male or female jellyfish, which grow into adults and produce jellyfish sperm and eggs. Then the cycle begins again.

Most jellyfish can reverse their development at most stages during this complicated life cycle. But once they grow into a sexually mature adult, they lose the ability to turn back the clock.

The immortal jellyfish disobeys this fundamental rule. Uniquely, even a sexually mature adult can revert to an immature polyp, "thus escaping death and achieving potential immortality". It's as if a butterfly suddenly reverted back into a caterpillar.

As with most cases of biological immortality, exactly how the immortal jellyfish pulls of this trick is a mystery. It seems to involve a bizarre reversed version of the cellular processes that go on during metamorphosis; the process by which juvenile caterpillars transform into adult butterflies.

Jellyfish don't have much in common with other animals, which is why their asexual reproductive strategy, and their immortality, seem so peculiar to our eyes.

Even in sexually reproducing animals, biological immortality isn't entirely unheard of

The two traits may actually be connected, says Bosch. If stem cells do play a vital role in animal biological immortality, then animals that have to carry potent stem cells in order to clone themselves might often be immortal.

On the other side of the coin, a reproductive strategy built around sex is almost invariably a one-way ticket to an early death.

"Maybe you could make the argument that you need lot of energy to make gametes [eggs and sperm], so that then kills the animal," says Bosch. That's certainly the case for male Antechinus, mice-like marsupials that almost literally mate themselves to death.

But even in sexually reproducing animals, biological immortality isn't entirely unheard of. The American lobster is a good example.

Most animals more or less stop growing when they reach sexual maturity, but not American lobsters. What's more, even as an adult this lobster can regrow a limb if it loses one by accident.

Shorter telomeres mean a shorter lifespan

Both of these features suggest American lobsters retain an impressive ability to regenerate, even into advanced adulthood. That might explain why large specimens are estimated to be at least 140 years old.

The lobsters' longevity may be connected to the behaviour of their DNA. The long chromosomes in animal cells have special tips on their ends, called telomeres, that help protect the DNA.

But whenever the cell divides and the chromosomes are replicated, the telomeres shorten a little bit, because the replication process can't quite reach to the very end of the chromosome.

Shorter telomeres mean a shorter lifespan. But American lobsters delay the inevitable using a telomere-lengthening enzyme called telomerase. A 1998 study revealed that this enzyme is found in all of their organs, where it presumably helps keep cells looking youthful for longer.

Telomerase enzymes appear to help tumours grow and spread

In other words, American lobster cells apparently don't age in a normal way, making the lobsters biologically immortal.

This telomere trick looks like a useful way for any organism to delay ageing. But there's actually very little evidence that the strategy is used, either by immortal plants or by immortal "lower" animals like the jellyfish. Bosch says it might be unique to "higher" animals.

Certainly, mammals also carry telomerases. In humans, they are active in HeLa cells: the first "immortal" human cells ever identified.

But in this case, the immortality is bad news. HeLa cells are so named because they were taken – without consent – from Henrietta Lacks, who died of cervical cancer in 1951.

Our "germ line" cells, which give rise to eggs and sperm, are ageless

Telomerase enzymes appear to help tumours grow and spread, which might be why mammals only use them in a few types of cell. HeLa cancer cells might be immortal, but their appearance cost Henrietta Lacks her life.

Cancer cells aren't the only immortal cells that can be found in the human body. Our "germ line" cells are ageless too. These are the cells that give rise to eggs and sperm, and it's vital that they can withstand ageing so that babies are born young.

The concept of young babies might sound like a tautology: surely all babies are young? But it's not, as Dolly the sheep demonstrates.

Dolly was cloned from sheep mammary gland cells, which aren't protected from ageing, and so she was born relatively "old". The telomeres in Dolly's cells were short even while she was a lamb, and she aged much more quickly than her non-cloned peers. Ultimately she was put down at the relatively tender age of six because of a lung disease.

There is perhaps a crumb of comfort here for anyone frightened of their own death

"The seed of immortality for organisms like us is that, every so often, we have a mechanism that can reset the clock," says Thomas.

At this point, it should come as no surprise that we don't know how the clock is reset in our germ cells. Telomerase enzymes are probably a factor, but they're not the full story. That means we're a long way from being able to reverse our own ageing, no matter what skin cream adverts might tell you.

Still, there is perhaps a crumb of comfort here for anyone frightened of their own death. We age as individuals, but because of the special properties of our germ cells, our lineage doesn't age. In that sense, humanity is immortal.