9-11 minutes - Source: BBC
"They're using Covid to defraud the American people," Mr Trump told delegates on the first day of the party convention in North Carolina.
Mr Trump repeated his much-disputed claims that mail-in ballots could lead to voter fraud.
Opinion polls suggest he currently trails Democratic challenger Joe Biden.
Addressing delegates in person at a party conference that has been dramatically scaled back by Covid-19, Mr Trump accused Democrats of "using Covid to steal an election".
"The only way they can take this election away from us is if this is a rigged election," he said. "We're going to win."
Mr Trump also warned of a "rigged" election in 2016, as he trailed Hillary Clinton in the polls.
On Monday, Mr Trump was officially nominated as a formality to be the Republican nominee at his party's convention in the city of Charlotte.
Supporters cheered him, chanting: "Four more years!"
The president is due to give a formal acceptance speech to the party jamboree on Thursday. It is unusual for candidates to address the convention before that point as Mr Trump has done.
Are mail-in ballots safe?Mr Trump has repeatedly asserted that expanded mail-in voting - which is expected to happen at an unprecedented level due the coronavirus pandemic - will lead to "the most corrupt election" in US history.
But there is scant evidence of widespread voter fraud, and very few examples of any related criminal prosecutions.
Ellen Weintraub, commissioner of the Federal Election Commission, has said: "There's simply no basis for the conspiracy theory that voting by mail causes fraud. None."
Mail-in voting is used by the US military, and even Mr Trump himself and members of his family.
But a recent slowdown in mail deliveries due to cost-saving measures in the US postal system has fuelled concerns that ballots might not be returned by election day.
Several states have sought to change their election laws to allow ballots to be counted days after the forthcoming presidential vote, which some analysts fear could lead to delays in declaring the presidential victor.
A primary election in New York City this June took weeks to determine a winner after poll officials were deluged with 10 times the normal number of mail-in ballots. There was no allegation of fraud, but the debacle raised fears of a protracted vote count this November.
Earlier this month, a New Jersey judge ordered a new vote after finding evidence of fraud in a May election that was conducted entirely by mail in Paterson. Four people were arrested, including a local city councilman and councilman-elect. The case has been frequently touted by the Trump campaign.
Republicans make case for TrumpThe first night of the Republican National Convention was a two-and-a-half hour rebuttal to the accusations Democrats levelled at Donald Trump during the four nights of their convention last week.
Did the president mishandle the coronavirus pandemic? The Republicans offered slick videos and first-hand accounts of the steps the president took to speed medical research, provide protective supplies and implement economic relief.
Is the president inflaming racial divisions in the US? Former football star Herschel Walker spoke of his 37-year friendship with Mr Trump. Tim Scott, the first black Republican senator since the late 19th Century, touted the president's work on sentencing reform and tax breaks for economically distressed communities.
Does the president lack empathy? Congressman Jim Jordan spoke of how Mr Trump offered sympathies when a relative died, and the president himself held a pre-taped White House forum where he offered words of support for coronavirus survivors and healthcare workers.
Polls suggest American voters have serious doubts about the president on all these issues - doubts that predated the Democratic convention attacks. Republicans have four days to assuage these concerns, chip away at Democratic opponent Joe Biden's lead and remind supporters what they like about Mr Trump's presidency.
It's an imposing task, but the Republicans have identified what work has to be done.
What else happened on the convention's opening night?Republicans said Monday night's theme was "Land of Promise" and pledged their convention would be less "negative" than the Democrats' conference last week.
But many of the Republican speakers warned in doom-laden tones what would happen if Americans voted in a President Biden.
Mr Trump's eldest son Donald Trump Jr said the Democratic presidential candidate was "basically the Loch Ness monster of the swamp".
Florida congressman Matt Gaetz warned of a "horror movie" if Democrats won the White House.
"They'll disarm you, empty the prisons, lock you in your home, and invite MS-13 to live next door," he said referring to a Central American immigrant street gang.
Charlie Kirk, who runs conservative student group Turning Point USA, told viewers: "Trump was elected to protect our families - our loved ones - from the vengeful mob that wishes to destroy our way of life, our neighbourhoods, schools, church and values."
Former UN Ambassador and South Carolina Governor Nikki Haley described facing discrimination as an Indian-American growing up in the South.
"I was a brown girl in a black and white world," she said.
But she described the "fashionable" idea that America is a racist country as a "lie".
How else was race addressed?After Mr Biden implied in last week's speech to his party convention that Mr Trump was a racist, the Republican conference sought to deflect that charge back on Democrats.
In a bid to woo a constituency that is crucial to Democratic electoral hopes, African-American speakers lavished praise on the president on Monday night.
The headline speaker was South Carolina Senator Tim Scott - the only black Republican senator. He said his grandfather had been forced out of school so he could pick cotton, but lived long enough to see his grandson elected.
"Our family went from cotton to Congress in one lifetime," he told delegates. He said America had made "tremendous progress" towards racial equality during the Trump presidency.
Former NFL star Herschel Walker, who is black, said: "It hurt my soul to hear the terrible names that people call Donald. The worst one is 'racist'.
"I take it as a personal insult that people would think I've had a 37-year friendship with a racist."
Georgia state representative Vernon Jones, a Democrat, accused his party of taking the black vote for granted.
"The Democratic Party does not want black people to leave the mental plantation they've had us on for decades," he said, evoking the legacy of slavery.
Kim Klacik, a black Republican candidate for Maryland's Baltimore district, accused Mr Biden of believing that black people "can't think for ourselves - that the colour of someone's skin dictates their political views".