Video report by ITV News Correspondent Emma Murphy
The Democratic candidate has promised to be a president for all Americans, regardless of who they voted for, after he defeated Donald Trump in the nail-biting race for the White House.
Mr Biden told crowds in Delaware: “I’m humbled by the trust and confidence you’ve placed in me.
“I pledge to be a president who seeks not to divide, but to unify. Who doesn’t see red and blue states, only sees the United States.
“And who will work with all my heart to win the confidence of all of you, and for that, I believe, is what America is all about.”
Mr Biden clinched victory after winning the key battleground of Pennsylvania on Saturday – some four days after polls closed – pushing him over the 270 electoral college votes threshold.
ITV News Correspondent Emma Murphy explains what will happen in the weeks before Joe Biden is sworn in as the 46th president of the US
The former vice president 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”.
It comes as a White House official announced that Donald Trump will “accept the results of a free and fair election”.
As the news broke on Saturday morning US time, many Americans took to the streets to express their joy after the knife-edge election.
In New York City, crowds gathered in the streets and in Times Square, banging pots and pans together in impromptu parties to celebrate Mr Biden’s win.
Car horns were sounded in the streets of many cities, including Los Angeles, while fireworks erupted in Atlanta, Georgia.
Jubilant demonstrators marched through Hollywood chanting “it’s a new day” as they celebrated Mr Biden’s election victory.
Activists carrying Black Lives Matter signs, Pride flags and placards bearing the names of people killed by police beat drums and danced along Sunset Boulevard as passing drivers in the liberal stronghold beeped their approval.
Prime Minister Boris Johnson congratulated Mr Biden and vice president-elect Kamala Harris, adding: “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.”
Mr Biden, who is expected to address the nation at around 1am UK time, said on Twitter: “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.”
In a statement, he said he is “honoured and humbled” by the trust the American people have placed in him and Ms 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,” he said.
Former president Barack Obama said he “could not be prouder” to congratulate Mr Biden and Ms Harris, while former Democratic presidential nominee Hillary Clinton hailed the pair as “a history-making ticket, a repudiation of Trump, and a new page for America”.
Her husband, former president Bill Clinton, posted: “America has spoken and democracy has won.
“Now we have a President-Elect and Vice President-Elect who will serve all of us and bring us all together.”
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.
In a tweet posted just hours before Mr Biden is set to address the nation – written entirely in capital letters – Mr Trump said he had won the election.
Twitter flagged this tweet pointing out that Mr Trump’s claim about election fraud is disputed.
On Saturday morning, a top election official dismissed Mr Trump’s claims of voter fraud.
Mr Trump posted a number of tweets earlier, alleging that “bad things” happened, referring to votes being “illegally received” and insisting he has won “by a lot”.
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.”
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”.
He later tweeted: “I WON THIS ELECTION, BY A LOT!”
Twitter flagged this tweet, saying official sources may not have called the race when it was tweeted.