Russian President Vladimir Putin has become one of the last world leaders to congratulate US President-elect Joe Biden on his victory.
The Kremlin had said it would wait for official results from November's vote before recognising the victory.
Democrat Biden won November's contest with 306 electoral college votes to the Republican incumbent's 232.
Mr Trump still refuses to concede, making unsubstantiated claims of widespread fraud.
Confirmation by the electoral college was one of the steps required for Mr Biden to take office.
In a blow to Mr Trump, one of his main Republican allies, Senate Majority Leader Mitch McConnell, congratulated Mr Biden.
Most world leaders congratulated Mr Biden within days of the 3 November poll, when it was clear he had defeated Mr Trump.
Mr Biden and Mr Putin had frosty relations when the former served as vice-president under Barack Obama.
Mexican President Andrés Manuel López Obrador has also finally congratulated Mr Biden.
Brazil's President, Jair Bolsonaro, has sent his "best wishes" to "President Joe Biden" and is ready to work with him, the foreign ministry said in a statement on Tuesday.
One leader who is not known to have congratulated Mr Biden is North Korea's Kim Jong-un.
What did Putin say in his message?
According to a Kremlin statement, Mr Putin wished Mr Biden every success and said he was "ready for collaboration and contacts with you".
He "expressed confidence that Russia and the United States, which have a special responsibility for global security and stability, could, despite their differences, really help to solve the many problems and challenges facing the world", the statement added.
But analysts say Mr Biden, who describes Mr Putin as an autocrat, is expected to take a tougher line on Russia than Mr Trump.
Russia has faced accusations of interference in the 2016 US election to help get Mr Trump elected.
In US-Russian relations, personal chemistry is so important. Mikhail Gorbachev and Ronald Reagan had it. Boris Yeltsin and Bill Clinton, too. Putin and Biden? Don't hold your breath.
On a trip to Moscow in his capacity as vice-president, Mr Biden concluded that Mr Putin had no soul. More recently he's labelled Russia the greatest threat to America. Doesn't sound promising, does it?
It doesn't help that Vladimir Putin waited so long to congratulate America's new leader. The Kremlin claimed it was doing things by the book, simply waiting for the official result. But the delay smacks of sour grapes.
There's little doubt the Russian authorities would have preferred Donald Trump to be re-elected. Not that US-Russian relations flourished under his administration but at least he avoided criticising Russia and its leader. Plus, the weakening of the Western alliance under President Trump was viewed positively by Russian officials.
Moscow is bracing itself for a tougher US approach to Russia and, possibly, additional sanctions. But whatever political, ideological or personal differences they may have, Presidents Biden and Putin will need to develop some kind of working relationship. There are issues of global importance - like arms control - where co-operation between Russia and America is vital.
While Mr Putin waited nearly a month and a half before congratulating Mr Biden, in 2016 he congratulated Mr Trump on his victory the day after the vote.
How did Biden greet confirmation of his win?
In a speech after the announcement, he said it was "time to turn the page". US democracy had been "pushed, tested and threatened", he said, but it had "proved to be resilient, true and strong".
He condemned Donald Trump's attempts to overturn the result, referring to the president's efforts to question the outcome and his legal challenges which have been rejected by courts across the country.
"Respecting the will of the people is at the heart of our democracy, even when we find those results hard to accept," Mr Biden said, speaking in Delaware.
"The flame of democracy was lit in this nation a long time ago," he added. "And we know that nothing not even a pandemic or an abuse of power can extinguish that flame."
He also warned that, with the coronavirus pandemic continuing to ravage the US, there would be difficult months ahead.
"There is urgent work in front of us," he said. "Getting this pandemic under control and getting the nation vaccinated against this virus."
He was speaking as the Covid death toll in the US - the country worst affected by the virus - rose above 300,000.
What happens next?
The results of the voting process will be sent to Washington DC and formally counted in a joint session of Congress on 6 January presided over by Vice-President Mike Pence.
That will pave the way for Joe Biden to be sworn in as president on 20 January.
On Tuesday, Mr Biden travels to Georgia to campaign for the Democrats in next month's Senate elections. Two seats will be decided on 5 January and could decide whether or not his party takes control of the chamber.