He said both China and Ukraine should look into Mr Biden, a Democratic presidential frontrunner, and his son.

"China should start an investigation into the Bidens," said Mr Trump.

The Democrats accuse Mr Trump of dangling military aid as a way to press Ukraine to dig dirt on Mr Biden.

Mr Biden responded by tweeting: "The idea of Donald Trump attacking anyone's integrity is a joke."

A 25 July phone call between Mr Trump and Ukrainian President Volodymyr Zelensky triggered an intelligence community whistleblower complaint and the impeachment inquiry.

But Mr Trump said on Thursday that the inquiry, which could culminate in a congressional attempt to remove him from office, is "crap".

In a late-night tweet, Mr Trump added that he had a duty to investigate alleged corruption "and that would include asking, or suggesting, other countries to help us out!"

What has Trump accused the Bidens of doing?

Mr Trump has accused Mr Biden and his son Hunter of corruption in their political and business dealings in Ukraine and China, without offering specific evidence.

When Hunter Biden joined Ukrainian natural gas company Burisma in 2014, questions were raised about a potential conflict of interest for his father.

Ukraine was undergoing a political transition after its pro-Russia president was forced out of office, while the elder Biden was the Obama administration's point man for the Eastern European country.

In 2016, Joe Biden pushed the Ukrainian government to fire its top prosecutor, Viktor Shokin, whose office had been scrutinising the oligarch owner of Burisma.

In a speech last year at a think tank, Mr Biden boasted of having forced Mr Shokin out by threatening to withhold a billion-dollar loan guarantee to Ukraine.

Mr Trump and his allies accuse Mr Biden of acting to protect his son. However, other Western officials and major financers of Ukraine's government also wanted Mr Shokin dismissed because he was seen as a barrier to anti-corruption efforts.

Last week, the Ukrainian prosecutor general who took over from Mr Shokin told the BBC there was no evidence of wrongdoing by Joe or Hunter Biden.

What did Trump say on Thursday?

When asked what Mr Trump sought as a "favour" from Ukrainian President Volodymyr Zelensky in this summer's phone call, Mr Trump responded: "Well, I would think that if they were honest about it, they'd start a major investigation into the Bidens. It's a very simple answer.

"They should investigate the Bidens," he said, speaking to reporters on the lawn of the White House.

"Likewise, China should start an investigation into the Bidens because what happened to China is just about as bad as what happened with Ukraine," he continued.

Mr Trump also suggested again without offering evidence that Mr Biden had "scammed" other countries and was responsible for China's "sweetheart" trade relationship with the US.

The head of the Federal Election Commission responded by tweeting a reminder that it is illegal for anyone to solicit anything of value from a foreign national in connection with a US election.

How have the Bidens responded?

A spokeswoman for Mr Biden's 2020 campaign accused Mr Trump of "flailing and melting down on national television, desperately clutching for conspiracy theories that have been debunked and dismissed by independent, credible news organisations".

Deputy Campaign Manager Kate Bedingfield's statement compared Mr Trump's latest comments to his "'Russia, if you're listening' moment from 2016 - a grotesque choice of lies over truth and self over the country".

The quote is a reference to Mr Trump's first election campaign, in which he urged Russia to find more than 30,000 emails deleted from his opponent Hillary Clinton's personal server.

A spokesman for Hunter Biden has told US media that Mr Trump's $1.5bn (£1.2bn) figure is false, and that the younger Biden did not receive any return or compensation from the Chinese firm.

What about the Bidens in China?

In 2013, then vice-president Mr Biden went to China on an official visit, where he met Chinese President Xi Jinping and other officials.

Hunter Biden and his daughter joined the vice-president, who had travelled with family members before.

During the two-day visit, Hunter met a Chinese banker, Jonathan Li, who would eventually become a business partner.

Mr Li founded a private equity fund shortly after the trip, and Hunter was on the board, although a spokesman for the younger Mr Biden told NBC News they did not discuss any business during the trip and the fund had been planned months earlier.

Hunter Biden was also not an equity owner in the fund during his father's term as vice-president, according to the spokesman.

Hunter has denied meeting any Chinese officials about the business. However, he reportedly helped arrange for Mr Li to shake hands with Joe Biden during his trip to Beijing, which stoked claims of influence-peddling.

This August, Republican Senator and Finance Chairman Chuck Grassley questioned Hunter's actions on the trip.

He said the younger Biden had a "history of investing in and collaborating with Chinese companies, including at least one posing significant national security concerns".

Brilliant strategy or political madness?

Once again Donald Trump is publicly stating something that has got him into hot water for saying privately.

It's either brilliant strategy or political madness. Or maybe a bit of both.

For those who believe that a president calling on a foreign government to investigate a political rival is an impeachable offence, Mr Trump has already offered more than enough evidence to proceed straight to a vote, no "inquiry" necessary.

If an abuse of presidential power is what elevates the foreign "request" to an impeachable act, then a bit more digging is required. Is there evidence that Mr Trump tied Ukrainian military aid to the Biden request? Has Biden come up at all during Mr Trump's trade negotiations with China?

Less than a minute before Mr Trump called on China to investigate the former vice-president, he spoke about those trade negotiations, noting that the US has "tremendous power" over China. How is that power being used?

Mr Trump is drawing a line in the sand. He says his Ukraine phone call was "perfect" - and, as if to prove it, he's saying and doing the same thing again and again.

If Democrats disagree, they can impeach him. And they just might.

How is the impeachment inquiry progressing?

Ex-US special envoy to Ukraine Kurt Volker has become the first witness to testify to Congress in the formal impeachment inquiry of President Trump.

Mr Volker was among those mentioned in an anonymous whistleblower's complaint about Mr Trump's call to the president of Ukraine.

The Republican foreign policy expert was addressing three Democratic-led House committees behind closed doors on Thursday.

Afterwards, Republicans and Democrats gave contrasting accounts of what Mr Volker's testimony meant for Mr Trump, with Republican Mark Meadows saying there was "no quid pro quo" and Democrat Eric Swalwell saying it showed there was a "shadow shakedown going on".

The whistleblower's complaint said that one day after the Trump-Zelensky call, Mr Volker and US ambassador to the EU, Gordon Sondland, met the Ukrainian president to advise him on how to "navigate" Mr Trump's request.

Mr Volker resigned from the Department of State last week after being named in the complaint.

Secretary of State Mike Pompeo had sought to block testimony from Mr Volker and four other Department of State officials whom Democrats wish to interview. 

In advance of his testimony, Mr Volker handed over documents to the committee, including text messages with Mr Trump's personal lawyer Rudy Giuliani.

On Friday morning Democratic congressional leaders published a letter setting out "grave concerns" over what they said was the Trump administration's attempt to "normalise the act of soliciting foreign powers to interfere in our elections".

They attached text message exchanges involving Mr Volker as well as an exchange between Bill Taylor, the top US diplomat in Ukraine, and Mr Sondland.

In one exchange, Mr Taylor tells Mr Sondland: "As I said on the phone, I think it's crazy to withhold security assistance for help with a political campaign."

Mr Sondland, a former Republican donor, replied: "Bill, I believe you are incorrect about President Trump's intention."

"The president has been crystal clear: no quid pro quos of any kind."

Mr Sondland then suggests "we stop the back and forth by text".

Quick facts on impeachment

Impeachment is the first part - the charges - of a two-stage political process by which Congress can remove a president from office

If the House of Representatives votes to pass articles of impeachment, the Senate is forced to hold a trial

A Senate vote requires a two-thirds majority to convict - unlikely in this case, given that Mr Trump's party controls the chamber

Only two US presidents in history - Bill Clinton and Andrew Johnson - have been impeached but neither was convicted and removed

President Nixon resigned before he could have been impeached

BBC