Video report by ITV News Washington Correspondent Robert Moore
The former vice-president was called as the winner of Pennsylvania on Saturday, meaning he has an insurmountable lead in the race for the White House.
The 77-year-old president-elect said he is “honoured” that America has “chosen me to lead our great country”, adding that it is time for the US to “unite” and “heal”.
Winning the vote in Wisconsin and Michigan – three states that voted for Mr Trump in 2016 – opened up several pathways to the presidency for the Democratic candidate.
But it was victory in Pennsylvania on Saturday, some four days after polls closed that saw him over the line.
Several states are still counting votes, and even if Mr Trump were to win all of them, he would still not amass enough support to topple Mr Biden, however, it looks like the majority of the remaining electoral college votes will go towards the Democrat.
Mr Trump, who has yet to concede, was playing golf at his course in Virginia when the election was called for Mr Biden.
Unlike in the UK where electoral districts declare the winner during an election, in the US, TV networks declare which way a state looks to have voted.
Each state elects members to the electoral college and they vote on December 14.
On January 6 the Senate then counts the votes and confirms who will be the next US president.
The president-elect will then be sworn in on January 20.
What happens now Joe Biden is president-elect?
Mr Biden, who is expected to address the nation at around 1am UK time on Sunday, said on Twitter: “America, I’m honoured that you have chosen me to lead our great country.
“The work ahead of us will be hard, but I promise you this: I will be a President for all Americans — whether you voted for me or not.
“I will keep the faith that you have placed in me.”
In a statement, Mr Biden said: “I am honoured and humbled by the trust the American people have placed in me and in Vice President-elect Harris.
“In the face of unprecedented obstacles, a record number of Americans voted. Proving once again, that democracy beats deep in the heart of America.
“With the campaign over, it’s time to put the anger and the harsh rhetoric behind us and come together as a nation.
“It’s time for America to unite. And to heal.
“We are the United States of America. And there’s nothing we can’t do, if we do it together.”
Shortly after Mr Biden's success was called, his running mate, Kamala Harris - who will become the first female and ethnic minority vice-president - tweeted a clip of her phoning him, telling the president-elect: "We did it. We did it Joe.
"You're going to be the next president of the United States."
Meanwhile, Mr Trump said in a statement he considers the election “far from over” as he repeated unsubstantiated claims of fraudulent ballots and vowed to press ahead with legal action.
He said: “We all know why Joe Biden is rushing to falsely pose as the winner and why his media allies are trying so hard to help him: they don’t want the truth to be exposed.
"The simple fact is this election is far from over.
“Joe Biden has not been certified as the winner of any states, let alone any of the highly contested states headed for mandatory recounts, or states where our campaign has valid and legitimate legal challenges that could determine the ultimate victor.
“In Pennsylvania, for example, our legal observers were not permitted meaningful access to watch the counting process.
"Legal votes decide who is president, not the news media.”
People in New York singing and celebrating Mr Biden's win
Mr Trump added: “Beginning (on) Monday, our campaign will start prosecuting our case in court to ensure election laws are fully upheld and the rightful winner is seated.
“The American people are entitled to an honest election: that means counting all legal ballots and not counting any illegal ballots.
“This is the only way to ensure the public has full confidence in our election.
“It remains shocking that the Biden campaign refuses to agree with this basic principle and wants ballots counted even if they are fraudulent, manufactured or cast by ineligible or deceased voters.
“Only a party engaged in wrongdoing would unlawfully keep observers out of the count room – and then fight in court to block their access.
“So what is Biden hiding? I will not rest until the American people have the honest vote count they deserve and that democracy demands.”
Listen to the ITV News US election podcast
On Saturday morning, a top election official dismissed Mr Trump’s claims of voter fraud.
Mr Trump posted a number of tweets on Saturday, alleging that “bad things” happened, referring to votes being “illegally received” and insisting he has won “by a lot”.
Democratic voters in Delaware, the state in which Mr Biden lives, reacted with joy after his win was announced
But Federal Election Commission commissioner Ellen Weintraub said there has been no evidence of voter fraud.
Speaking on CNN after Mr Trump’s string of tweets – which were flagged by Twitter as containing information about the election that may be “misleading” – Democrat Ms Weintraub said: “State and local officials, and poll workers throughout the country, really stepped up.
“And there have been very few complaints about how this election has run.
“Very few substantiated complaints, let me put it that way. There is no evidence of any kind of voter fraud.
“There is no evidence of illegal votes being cast.
“In fact, and you don’t have to take my word for it because people throughout the country, non-partisan election experts have come out and hailed this election and how it was run.
“If you want to look at the state of Pennsylvania, which the president seems to be focused on, Senator (Pat) Toomey has come out and said he has seen no evidence of fraud there.
“The Republican leader of the state senate there has also said he has seen no evidence of fraud.”
She added: “There really has been no evidence of fraud. None of the complaints have attached any evidence of fraud.
“Really, we should feel very proud of ourselves.”
Her comments came after Mr Trump wrote on Twitter that tens of thousands of votes were “illegally received” after 8pm on Tuesday “totally and easily changing the results in Pennsylvania and certain other razor thin states”.
A record 102 million Americans voted early or via absentee as the coronavirus pandemic continues to grip the country.
Soon after the news of Mr Biden and Ms Harris's victory, congratulations began pouring in from across the world.
Prime Minister Boris Johnson tweeted: “Congratulations to Joe Biden on his election as President of the United States and to Kamala Harris on her historic achievement.
“The US is our most important ally and I look forward to working closely together on our shared priorities, from climate change to trade and security.”
Foreign Secretary Dominic Raab adopted a conciliatory tone when he congratulated both Mr Biden and Ms Harris, but added that Mr Trump "fought hard in what proved a close contest".
He continued: “We're looking forward to working with the new administration on all of our shared interests, from tackling Covid-19 to counter-terrorism, and collaborating closely through our Presidencies of COP26 and the G7 next year.
"The friendship between the UK and US has always been a force for good in the world."
Former Democratic president Barack Obama said he “could not be prouder” to congratulate President-elect Joe Biden and Vice President-elect Kamala Harris, saying they will have won “a historic and decisive victory” once all votes are counted.
“We’re fortunate that Joe’s got what it takes to be President and already carries himself that way,” he said.
“Because when he walks into the White House in January, he’ll face a series of extraordinary challenges no incoming President ever has – a raging pandemic, an unequal economy and justice system, a democracy at risk, and a climate in peril.
“I know he’ll do the job with the best interests of every American at heart, whether or not he had their vote.”
ITV News Correspondent Emma Murphy on Joe Biden, the person and the politician
Joe Biden's presidential win culminates nearly 50 years in politics. He was elected a senator at 30 years of age, only for his world to fall apart, five weeks later, when his wife and daughter were killed in a car crash. One of his surviving sons, Beau Biden, also died young - at 46 - of brain cancer. Mr Biden tried and failed twice before to secure the Democratic Party nomination for President. He made it, of course, into the White House as Barack Obama's vice president. Now he's due to become the commander-in-chief himself.