The Trump administration has released details of a phone conversation in July that has triggered a US impeachment inquiry against the president.
According to the notes, Donald Trump asked the Ukrainian leader to look into corruption claims involving the son of Joe Biden, Mr Trump's possible rival in next year's presidential election.
Concerns about the call were initially raised by a whistleblower.
The Democrats accused Mr Trump of seeking foreign help to smear a rival.
Under the US constitution, a president can be impeached for "treason, bribery, or other high crimes and misdemeanours" - a procedure that can lead to removal from office.
In July, Mr Trump froze military aid to Ukraine but he has insisted that this was not used to put pressure on the new government in Kiev.
What does Trump say about Biden in the call?
Mr Trump discusses with his newly elected Ukrainian counterpart, Volodymyr Zelensky, the 2016 removal of a prosecutor, Viktor Shokin, according to notes of their 25 July telephone conversation released by the White House.
The US president is quoted as saying in the half-hour call: "I heard you had a prosecutor who was very good and he was shut down and that's really unfair.
"A lot of people are talking about that, the way they shut your very good prosecutor down and you had some very bad people involved."
He continues: "The other thing, there's a lot of talk about Biden's son, that Biden stopped the prosecution [of Mr Biden's son] and a lot of people want to find out about that so whatever you can do with the [US] Attorney General would be great.
"Biden went around bragging that he stopped the prosecution so if you can look into it... It sounds horrible to me."
Mr Zelensky says in response: "We will take care of that and we will work on the investigation of the case."
Thanking Mr Trump, Mr Zelensky says he stayed in Trump Tower in New York City during a previous visit to the US.
On the call, the US president also asks Mr Zelensky to work with US Attorney General William Barr and Mr Trump's personal lawyer, Rudolph Giuliani, to look into the matter, according to the notes.
The Department of Justice said on Wednesday that Mr Trump had not spoken with the attorney general about having Ukraine investigate Mr Biden, and Mr Barr had not communicated with Ukraine.
What is the claim against Joe Biden?
Mr Trump and his conservative allies have focused on how Mr Biden, as US vice-president in 2016, lobbied Ukraine to fire Mr Shokin.
Mr Shokin's office had opened an investigation into Burisma, a natural gas company on which Mr Biden's son, Hunter Biden, was a board member.
Other Western officials had called for Mr Shokin to be fired because of the perception that he was soft on corruption.
Mr Biden last year told a foreign policy event how he threatened to withhold $1bn in aid to Ukraine unless Mr Shokin was removed.
No evidence has come to light so far of wrongdoing by the Bidens.
How the controversy unfolded
- 18 July - President Trump orders White House aide to hold back almost $400m in military aid to Ukraine, report US media
- 25 July - President Trump speaks with Ukraine's leader in a 30-minute phone call
- 9 September - Congress learns of a whistleblower's complaint about the call, but is blocked by the Trump administration from viewing it
- 11 September - Military aid for Ukraine is cleared for release by the Pentagon and US Department of State
- 23 September - Trump confirms he withheld Ukrainian aid, saying it was due to concerns about "corruption"
- 24 September - Trump says the aid was withheld so that other countries would pay more
What is the background of the call?
Mr Trump had promised that a "complete, fully declassified and unredacted transcript" of the 25 July call would be released.
But the details disclosed by the White House on Wednesday morning were notes of the conversation taken by US officials who listened in.
The July call occurred days after Mr Trump directed the US government to withhold about $391m (£316m) in military aid to Ukraine.
In the summary of the call, Mr Trump does not tell the Ukrainian president that US aid hinges upon him investigating Mr Biden.
But Mr Trump stresses the importance of US financial support, before pressing Mr Zelensky for action on the Democratic candidate.
"I will say that we do a lot for Ukraine," Mr Trump says.
What's the reference to 'CrowdStrike'?
In one of the most cryptic passages from the call, Mr Trump mentions a "server" and CrowdStrike.
He asks Mr Zelensky to "do us a favour" adding: "I would like you to find out what happened with this whole situation with Ukraine, they say CrowdStrike... I guess you have one of your wealthy people... The server, they say Ukraine has it."
CrowdStrike is a cyber-security firm that was hired by the Democratic National Committee (DNC) to investigate the 2016 hack of the party's email server.
That cyber-breach led to the leak of numerous emails that embarrassed Hillary Clinton's presidential campaign.
CrowdStrike determined a pair of hacking groups with suspected Russian ties was responsible for the hack.
Mr Trump has previously questioned why the DNC did not turn over the hacked email server to the FBI, instead of asking CrowdStrike to investigate.
The US president told the Associated Press news agency in April 2017 that he had "heard" CrowdStrike was "owned by a very rich Ukranian".
In the phone call with Mr Zelensky, Mr Trump appears to suggest the DNC server is stored somewhere in Ukraine.
What's the reaction?
On Wednesday morning at the United Nations General Assembly in New York City, Mr Trump said it was the "single greatest witch hunt in American history".
"The way you had that built up that call, it was going to be the call from hell," said Mr Trump, who is up for re-election in November 2020.
"It turned out to be a nothing call."
Speaking alongside Mr Trump at the event, Mr Zelensky told reporters: "You heard that we had a, I think, good phone call.
"It was normal. We spoke about many things, so I think you read it that nobody pushed me."
"In other words, no pressure," interjected Mr Trump.
But California Democrat Adam Schiff, the Democratic chairman of the House Intelligence Committee, told reporters the phone call shows "a classic, mafia-like shakedown of a foreign leader".
What's happening with the impeachment inquiry?
The Democratic leader of the House of Representatives, Speaker Nancy Pelosi, on Tuesday threw her weight behind an official impeachment inquiry into the president's actions.
A House vote to impeach the president could trigger a trial in the Senate on whether to remove Mr Trump from office.
Congress' probe focuses partly on whether Trump abused his presidential powers and sought to help his own re-election by seeking the aid of a foreign government to undermine Mr Biden.
Mrs Pelosi said such actions would mark a "betrayal of his oath of office" and declared: "No one is above the law."
How did the controversy arise?
The inspector general for the intelligence community wrote to the Director of National Intelligence in August about the Trump-Zelenksy call.
A whistleblower from within the US intelligence community had filed a complaint about the matter.
Federal law requires such complaints to be disclosed to Congress, but the Trump administration has so far refused to do so.
The US Department of Justice says the whistleblower had heard the information from "White House officials", and did not have first-hand knowledge of the call.
The department's spokeswoman, Kerri Kupec, said on Wednesday it had reviewed a record of the phone call, and determined "there was no campaign finance violation and that no further action was required".