Here is a selection of photos that span their lives.

The early years

Born in the wake of World War Two, in June 1946, Donald John Trump was the fourth child of New York real estate tycoon Fred Trump and Mary Anne MacLeod Trump. Despite the family's wealth, he was expected to do the most menial jobs within his father's company and was sent to a military academy at age 13 after he started misbehaving in school.

He attended the University of Pennsylvania and became the favourite to succeed his father in the family business after his older brother, Fred, opted to become a pilot.

Joseph Robinette Biden Jr was born in Scranton, Pennsylvania in 1942. He was the first of four children, in an Irish-American Catholic family. Young Joe's biggest challenge was overcoming a speech impediment - a stutter - that afflicted him well into high school. His technique of practising speaking in front of a mirror paid off after several months.

Mr Biden attended the University of Delaware and then law school at Syracuse University.

He later married his first wife, Neilia, and started his political career in Wilmington.

The 1970s

Mr Trump says he got into the property business with a "small" $1m loan from his father, before joining Fred Trump's company. There, he helped manage an extensive portfolio of residential housing estates in New York City, eventually taking control of the company. In 1971, he renamed it the Trump Organization.

Six years later, Donald Trump married his first wife, Ivana Zelnickova, a Czech athlete and model. His children from his first marriage - Donald Jr, Ivanka and Eric - now help run Trump Organization, though he is still chief executive.

Joe Biden was eagerly waiting to take up his seat in the US Senate, having been elected in 1972, when tragedy struck. His wife and infant daughter Naomi were killed in a car accident. His sons Beau and Hunter were seriously injured.

Mr Biden famously took the oath of office for his first term as a Democratic Party senator from the hospital room of his toddler sons.

The 1980s

In the late 1970s Mr Trump stepped his ambitions up a gear, shifting his property focus from Brooklyn and Queens to glitzy Manhattan. After snapping up a rundown hotel and transforming it into the Grand Hyatt, he built the most famous Trump property - Trump Tower on Fifth Avenue. It opened in 1983.

Other properties bearing the famous name followed - Trump Place, Trump World Tower, Trump International Hotel and Tower - and his powerful brand began to draw media interest.

But not everything he touched turned to gold. Mr Trump's ventures have led to four business bankruptcy filings.

During his first 14 years in Washington, Mr Biden rebuilt his personal life after the deaths of his wife and daughter. He committed to giving his sons a semblance of a normal life, and commuted each day from the family home in Delaware to Washington DC. He eventually remarried, to schoolteacher Jill Jacobs, with whom he had another child, Ashley.

Mr Biden established himself on the Senate Judiciary Committee, and began to build a national profile. In 1987, he launched his first go at the US presidency, but withdrew after he was accused of plagiarising a speech by the then leader of the British Labour Party, Neil Kinnock.

The 1990s

Property alone was not enough for Mr Trump, who moved into the entertainment sector, snapping up a clutch of beauty pageants in 1996: Miss Universe, Miss USA, and Miss Teen USA. In his personal life, after splitting with Ivana he married actress Marla Maples in 1993.

They had a daughter, Tiffany, before divorcing in 1999 - the same year Mr Trump's father died.

"My father was my inspiration," Mr Trump said at the time.

On 11 October 1991, the US public were glued to their TVs as Anita Hill, a law professor at the University of Oklahoma, testified before the Senate Judiciary Committee. The committee was holding a hearing into the nomination for the US Supreme Court of Clarence Thomas. Ms Hill alleged he had sexually harassed her on many occasions when they had both worked for the Reagan administration.

As chairman of the committee, Joe Biden led the hearing. His handling of Ms Hill's evidence has long been criticised.

The hearing was conducted by an all-white, all-male panel, and several women apparently willing to back up Ms Hill's account were not called by Mr Biden to testify.

Speaking in a TV interview in April 2019, Mr Biden said that he was "sorry for the way she got treated".

The 2000s

In 2003, Mr Trump fronted a new reality TV show that played to his reputations as both a businessman and a media personality. Called The Apprentice, the programme featured contestants competing for a shot at a management job in Mr Trump's commercial empire.

He hosted the show for 14 seasons, and claimed in a financial disclosure form that he had been paid a total of $213m by the network during the show's run.

Meanwhile, in 2005, he married his current wife, Melania Knauss, a Yugoslavian-born model. The couple have one son, Barron William Trump.

Mr Biden had another shot at the presidency in 2008 before dropping out. But while his campaign had failed to break through, he was to reappear later that year in a role that assured him international prominence. On 23 August 2008, Mr Obama introduced Joe Biden as his vice-presidential running mate.

It was a winning ticket and the pair eventually served two terms, establishing a close working relationship in which Mr Biden frequently called Mr Obama his "brother".

The 2010s

It was not until June 2015 that Mr Trump formally announced his entrance into the race for the White House. His campaign for the presidency was rocked by controversies, including the emergence of a recording from 2005 of him making lewd remarks about women, and claims, including from members of his own party, that he was not fit for office.

But he consistently told his army of supporters that he would defy the opinion polls, which mostly had him trailing his Democratic rival Hillary Clinton. He said his presidency would strike a blow against the political establishment and "drain the swamp" in Washington.

He took inspiration from the successful campaign to get Britain out of the European Union, saying he would pull off "Brexit times 10". Despite almost all the predictions, Mr Trump was victorious in the 2016 election. He was inaugurated as the 45th US president on 20 January 2017.

In a surprise ceremony in the final days of his presidency, Mr Obama awarded Mr Biden the Presidential Medal of Freedom - the nation's highest civilian honour.

"To know Joe Biden is to know love without pretence, services without self-regard and to live life fully," the then president said.

It had been a successful partnership, but a period not without trauma for Mr Biden, whose son Beau died of brain cancer in 2015 at the age of 46. The younger Biden was seen as a rising star of US politics and had intended to run for Delaware state governor in 2016.

2020

Mr Trump's re-election campaign has been conducted against the backdrop of the coronavirus pandemic, in which 230,000 Americans have died, and seen the president himself become infected. First Lady Melania Trump and their son Barron caught the virus too, along with a number of staff at the White House.

In the days before the election on 3 November, Mr Trump urged states to shun lockdowns, while continuing his schedule of rallies in battleground states.

The two presidential rivals' divisions over the coronavirus have been deep, with Mr Biden having said the president's handling of the worsening coronavirus crisis was an "insult" to its victims.

"Even if I win, it's going to take a lot of hard work to end this pandemic," he said. "I do promise this - we will start on day one doing the right things."

More than 90 million Americans have voted early, many of them by post, in a record-breaking voting surge driven by the pandemic.