Joe Biden has said it is "time to heal" the US in his first speech as president-elect, vowing "not to divide but to unify" the country.
"Let's give each other a chance," he said at an event in Delaware addressing those who did not vote for him.
Mr Biden defeated incumbent President Donald Trump following a cliff-hanger vote count after Tuesday's election.
Mr Trump has yet to concede and has not spoken publicly since his defeat was announced while he was playing golf.
The result makes Mr Trump the first one-term president since the 1990s. His campaign has filed a barrage of lawsuits in various states but election officials say there is no evidence that the vote was rigged against him, as he has said.
Spontaneous celebrations erupted in major cities after media outlets announced Mr Biden's victory on Saturday. Disappointed Trump supporters demonstrated in some cities but there were no reports of incidents.
Biden: 'We have to stop treating our opponents as enemies'
Addressing cheering supporters in a car park in his hometown of Wilmington, Mr Biden said: "I pledge to be a president who seeks not to divide, but to unify; who doesn't see red states and blue states, only sees the United States."
Mr Biden - who has won more than 74 million votes so far, the most ever for a US presidential candidate - hailed the "diverse" support he gathered during the campaign, and thanked African-American voters in particular.
But he also reached out to Trump supporters directly.
"It's time to put away the harsh rhetoric, lower the temperature, see each other again, listen to each other again," Mr Biden said, without mentioning his rival in the election. "And to make progress, we have to stop treating our opponents as enemies."
The president-elect, who arrived on stage wearing a face mask, announced that he would form his coronavirus response committee to ensure it is ready to implement decisions from his inauguration day in January, because "that's the only way we can get back to living".
The Trump administration's response to the pandemic was at the centre of the presidential campaign, and drew heavy criticism from Mr Biden, who said his plan would be "built on bedrock science". The US has reported more than 237,000 deaths, the most of any country.
Harris: 'You chose hope and unity'
Mr Biden was introduced by his running mate, Kamala Harris, who is about to become the first female vice-president in the country's history.
She will also be the first black and first Asian-American vice-president when she takes office, following a year that has seen nationwide protests on the issue of racial injustice.
Ms Harris paid tribute to her late mother, an Indian immigrant to the US, and the other generations of black, Asian, white, Latina, and native women who had "paved the way for this moment tonight".
"While I may be the first woman in this office, I will not be the last," she added.
Reflecting on the election campaign, she told supporters: "When our very democracy was on the ballot in this election, the very soul of America at stake and with the whole world watching, you ushered in a new day for America."
She added: "You chose hope and unity, decency, science and yes, truth - you chose Joe Biden as the next president of the United States. And the road ahead will not be easy but America is ready, and so are Joe and I."
A change in presidential rhetoric
In an earlier time, Joe Biden's victory speech would have seemed fairly routine. The president-elect spoke of unity, of ending acrimony, of the potential and power of the American people. Those are sentiments many a winning politician have touched on in the past.
Coming on the heels of Donald Trump's presidency, however, they mark a sharp contrast. The president Mr Biden will replace was one who was often criticised for stoking cultural divisions and doing little to tamp down the unrest that broke out in many US cities earlier this year.
Mr Biden spoke of his election being an "inflection point" that would allow the American people to make a decision about "who we are and who we want to be".
As Saturday night's speech demonstrated, at the very least, the Biden presidency will mark a change in presidential rhetoric. The American people have a president-elect who talks about bringing the country together; about being a leader for all the people. Saying it is the easy part; now he has to do it.
Trump 'not planning to concede'
The BBC projected Mr Biden's victory on Saturday, after gains in the key battlegrounds of Pennsylvania and Nevada propelled him over the 270 electoral college vote threshold required to clinch the White House.
This prediction is based on the unofficial results from states that have already finished counting their votes, and the expected results from states like Wisconsin where the count is continuing.
Mr Trump has not spoken in public since the numbers were announced, but he repeated previous claims of voter fraud in a tweet, which Twitter soon marked as a "disputed" claim. The Trump campaign has indicated their candidate does not plan to concede.
After Mr Biden was projected to win Mr Trump remained defiant, saying Mr Biden was "falsely posing as the winner" and insisting the election was "far from over". The president took more than 70 million votes, the second-highest tally in history.
The response from senior Republicans has been muted. Republican Party Chairwoman Ronna McDaniel tweeted that "the media doesn't decide who wins elections, voters do".
But Senator Mitt Romney, a critic of Mr Trump, congratulated Mr Biden and Ms Harris. He said he and his wife "know both of them as people of good will and admirable character".
What happens now
Mr Trump has vowed to contest the election results on several fronts. A recount will be held in Georgia, where the margins are tight, and Mr Trump wants the same in Wisconsin. He has also vowed to take legal action to the Supreme Court, alleging voting fraud without evidence.
If the election result is challenged, it would require legal teams to challenge this in the state courts. State judges would then need to uphold the challenge and order a recount, and Supreme Court justices could then be asked to overturn a ruling.
On Saturday, the Trump campaign filed a lawsuit over ballots cast on election day in Arizona that it claims were incorrectly rejected. Arizona's secretary of state, however, said in a statement that the case was "grasping at straws".
Meanwhile, votes in some states are continuing to be counted and results are never official until final certification, which occurs in each state in the weeks following the election.
This must be done before 538 chosen officials (electors) from the Electoral College - which officially decides who wins the election - meet in their state capitals to vote on 14 December.
The electors' votes usually mirror the popular vote in each state. However, in some states this is not a formal requirement.
The new president is officially sworn into office on 20 January after a transition period to give them time to appoint cabinet ministers and make plans.
The handover of power takes place at a ceremony known as the inauguration, which is held on the steps of the Capitol building in Washington DC. After the ceremony, the new president makes their way to the White House to begin their four-year term in office.
How election is third time lucky for Biden
Mr Biden, who will be 78 when he takes office, making him the oldest first-term president in American history, ran for the White House twice before.
In 1988 he withdrew from the race after he admitted to plagiarising a speech by the then leader of the British Labour Party, Neil Kinnock.
In 2008, he tried again to get the Democratic nomination before dropping out and joining Barack Obama's ticket.
His eight years as vice-president allowed him to lay claim to much of Mr Obama's legacy, including passage of the Affordable Care Act.
The six-term senator from Delaware was first elected in 1972.
Early in his career, he sided with southern segregationists in opposing court-ordered school bussing to racially integrate public schools.
He was also a fierce advocate of a 1994 anti-crime bill that many on the left now say encouraged lengthy sentences and mass incarceration.
Most Americans know that Joe Biden's life has been marked by personal tragedy - experiences he often refers to.
In 1972, he lost his first wife, Neilia, and baby daughter, Naomi, in a car accident. He famously took the oath of office for his first Senate term from the hospital room of his toddler sons Beau and Hunter, who both survived the accident.
In 2015, Beau died of brain cancer at the age of 46, and Joe Biden said this played a role in his decision not to run for president in 2016.