European leaders have expressed sadness at the UK leaving the EU, with France's Emmanuel Macron emphasising Britain's "unrivalled ties" with the French.
Mr Macron said he was "deeply sad" while the EU's Guy Verhofstadt pledged to try and "ensure the EU is a project you'll want to be a part of again".
Celebrations and anti-Brexit protests were held on Friday night to mark the UK's departure.
Ex-Brexit Secretary David Davis said everyone would be a winner in the end.
The UK officially left the European Union on Friday at 23:00 GMT after 47 years of membership, and more than three years after it voted to do so in a referendum.
Brexit parties were held in some pubs and social clubs as well as in London's Parliament Square, as the country counted down to its official departure.
In Scotland, which voted to stay in the EU, candlelit vigils and anti-Brexit rallies were held.
In a message released on social media an hour before the UK left, Prime Minister Boris Johnson vowed to bring the country together and "take us forward".
"For many people this is an astonishing moment of hope, a moment they thought would never come," he said. "And there are many of course who feel a sense of anxiety and loss."
'Not leaving Europe'
In an open letter to the British public, French President Mr Macron said he was thinking of the millions of Britons "who still feel deeply attached to the European Union".
"You are leaving the European Union but you are not leaving Europe," he said. "Nor are you becoming detached from France or the friendship of its people.
"The Channel has never managed to separate our destinies; Brexit will not do so, either."
Mr Macron also said the EU must learn lessons from the "shock" of Brexit, adding: "I am convinced therefore that Europe needs new momentum."
And he defended the way France acted in the Brexit negotiations, saying neither the French nor anyone else in the EU was "driven by a desire for revenge or punishment".
Meanwhile, the EU Parliament's Brexit co-ordinator Mr Verhofstadt responded to a message which had been projected onto the White Cliffs of Dover by a pro-EU group.
"We will look after your star and work to ensure the EU is a project you'll want to be a part of again soon," he said.
Speaking on BBC Radio 4's Today programme, Mr Davis - who quit as Brexit secretary in protest at former prime minister Theresa May's Brexit plan - said it would be a "fair race" to reach a trade deal with the EU by the end of 2020 but "it can be done".
The UK is aiming to sign a permanent free trade agreement with the EU, along the lines of the one the EU has with Canada, by the end of the transition period in December.
Mr Davis said reaching a deal was "not a charitable exercise, this is an exercise of both sides recognising their own best interests".
European leaders have warned that the UK faces a tough battle to get a deal by that deadline.
Mairead McGuinness, the vice president of the European Parliament, said progress to agree a trade deal "might be left to the very last minute".
"Normally in trade negotiations we're trying to come together," she told BBC Breakfast. "For the first time we're going try and negotiate a trade agreement where somebody wants to pull away from us. I can't get my head around that and I think it's going to be quite complicated."
We are separate after more than 40 years, but remember much of the status quo will hold for now - the UK and the EU, the awkward couple, finally divorced - but still sharing a house and the bills.
But what the prime minister hails as a new era, a bright new dawn, starts months of hard bargaining with our neighbours across the Channel.
Labour leadership hopeful Emily Thornberry said the exit talks were unlikely to go smoothly and said she expected the country would be "back in no-deal territory by the summer".
The shadow foreign secretary, speaking at an event in Bristol featuring the four Labour leadership candidates, said her party would need a Remain-backing leader who had been "on the right side of the argument all along".
However, the other three candidates - Keir Starmer, Rebecca Long-Bailey and Lisa Nandy - said the party needed to move on from debates over Brexit.
Shadow business secretary Ms Long-Bailey said that Labour needed to make sure Boris Johnson negotiated the "best possible trade deal" that could help "rebuild our communities".
Whilst never the most enthusiastic member, the UK was part of the European project for almost half a century.
On a personal level, EU leaders tell me they'll miss having the British sense of humour and no-nonsense attitude at their table.
If they were to be brutally honest they'd have admitted they'll mourn the loss of our not-insignificant contribution to the EU budget too.
But now we've left the "European family" (as Brussels insiders sometimes like to call the EU) and as trade talks begin, how long will it take for warm words to turn into gritted teeth?
UK citizens will notice few immediate changes now that the country is no longer in the European Union.
Most EU laws will continue to be in force - including the free movement of people - until 31 December, when the transition period comes to an end.
Celebrations and protests
Thousands gathered in Parliament Square to celebrate Brexit on Friday night, singing patriotic songs and cheering speeches from leading Brexiteers, including Nigel Farage.
The Brexit Party leader said: "This is the greatest moment in the modern history of our great nation."
Pro-EU demonstrators earlier staged a march in Whitehall to bid a "fond farewell" to the union.
Police in Whitehall arrested four men and also charged one man with criminal damage and being drunk and disorderly, while in Glasgow one man was arrested.
Meanwhile, other symbolic moments on a day of mixed emotions included:
- The Union flag being removed from the European Union institutions in Brussels
- The Cabinet meeting in Sunderland, the first city to declare in favour of Brexit when the 2016 results were announced
- A light show illuminating 10 Downing Street and Union flags lining The Mall
- A 50p coin to mark the occasion entering circulation
- The building of the UK government's delegation to the EU changed its name and sign