China has finally congratulated Joe Biden on his projected win in the US presidential election, breaking a frosty period of silence.

"We respect the choice of the American people. We extend congratulations to Mr Biden and Ms Harris," a foreign ministry spokesman said on Friday.

The China-US relationship is crucial to both sides, and the wider world. Tensions have soared in recent times, over trade, espionage and the pandemic.

Russia is yet to offer well wishes.

Four years ago, the Russian leader Vladimir Putin was among the first to congratulate Donald Trump on his election victory, but there has been no tweet, telegram or phone call to Mr Biden this time.

"We believe the correct thing to do would be to wait for the official election result," Kremlin spokesman Dmitry Peskov told reporters.

Mr Biden has taken calls from a string of global leaders in recent days. Congratulations began to pour in from Saturday, when US networks projected the result and he declared victory.

President Trump has continued to make unsubstantiated allegations of voter fraud without providing clear evidence. On Thursday, a group of federal and state election officials directly rebutted such claims, saying the vote was "the most secure in American history".
What does China's message tell us?

To put it bluntly, it shows China's leaders - and specifically Xi Jinping, the country's powerful president - have accepted the result and expect Joe Biden to be inaugurated as president in January despite legal challenges from the Trump campaign.

Until this point, the Chinese government had been holding out, saying only that it had "noticed that Mr Biden had been declared winner".

After offering the congratulations on Thursday, foreign ministry spokesman Wang Wenbin added: "We understand the results of the US election will be determined according to US laws and procedures."

The relationship between China and the US - the world's two biggest economies - is hugely consequential. A trade war initiated by President Trump has damaged relations, as has his insistence on often calling Covid-19, the "Chinese virus" or "kung flu".

The two countries have also sparred over espionage, China's military build-up in the South China Sea and the mass detention of Muslims in western China.

Who China preferred to see in the White House was a matter for debate before the election. The US intelligence community believed Xi Jinping and the Communist Party leaned towards Mr Biden.

But Professor Yan Xuetong, at Beijing's Tsinghua University, said China would rather have seen four more years of Donald Trump.

"Not because Trump will do less damage to China's interests than Biden, but because he definitely will damage the US more than Biden," he told the BBC before the election.

It's unclear how much Joe Biden would shift US policy towards China, as there is rare cross-party agreement in Congress on getting tough with Beijing over trade and other issues.

Experts predict a tactical shift, however - they see a Biden White House working more with allies to confront China, while also seeking to co-operate in other areas, like fighting climate change.
Which other countries are we waiting for?
Russia

The BBC's Steve Rosenberg in Moscow says there is a suspicion that the lack of congratulations reflects the fact that Moscow is not excited by the outcome.

Mr Biden is a vocal critic of Moscow and recently identified Russia as the biggest threat to America.

Mr Trump has rarely criticised Russia or Mr Putin, and Russia was accused of interfering in the 2016 election to get Mr Trump elected.
Brazil

President Jair Bolsonaro is often considered to be an ally of Mr Trump, so much so that he has been described as the "Trump of the Tropics".

The Brazilian leader's failure so far to congratulate Mr Biden, therefore, comes as no surprise.

He has crossed swords with the former vice-president in the past, describing his call during an election debate for the US to push Brazil to protect the Amazon rainforest better as "disastrous and unnecessary".

Brazilian media reports quoting government sources suggest Mr Bolsonaro plans to wait until Mr Trump's legal challenges are completed.

On Thursday, when he was asked about the issue, the Brazilian leader responded: "But has it finished, have the elections already finished?"
Mexico

Andrés Manuel López Obrador is another Latin American leader who has tried to maintain good relations with Mr Trump, despite tensions over the US president's hardline policy on migrants and in particular his pledge to build a wall along the two countries' border.

The Mexican leader has therefore been cautious about the US election, and on Sunday said he would wait for "all legal issues" to be resolved.

"We don't want to be imprudent. We don't want to act lightly," he said at a news conference, adding that Mexico had a "very good relationship" with both candidates.

Mr López Obrador's equivocal position has drawn criticism from several senior US Democrats, with Texas Congressman Joaquin Castro describing it as "stunning diplomatic failure".
North Korea

There has been no response to the election result from Supreme Leader Kim Jong-un or from the North Korean media - which is all state-controlled.

In 2016, North Korea acknowledged Donald Trump's victory within a few days of his election.

Mr Trump and Mr Kim have had a stormy relationship, though they have maintained contact through three historic face-to-face meetings.

Mr Biden describes Mr Kim as a thug and says he is not interested in any personal diplomacy with him. The North Korean leader has called Mr Biden "a fool of low IQ".