The Democrats' lawyer, Daniel Goldman, asks Vindman about text messages between Kurt Volker, the former US special envoy to Ukraine, and a senior adviser to President Zelensky.

Goldman notes that a potentially problematic message from Volker to the Ukrainian presidential aide took place less than half an hour before the 25 July call.

Volker wrote: "Heard from the White House - assuming President Z convinces Trump he will investigate / 'get to the bottom of what happened' in 2016, we will nail down date for visit to Washington."

The BBC's Anthony Zurcher calls this "the clearest acknowledgement of a quid pro quo" - a latin term meaning an exchange of favours, and a smoking gun sought by impeachment investigators.

Read more on the text messages here.

Daniel Goldman, the lead Democratic lawyer for the impeachment hearings, asks Vindman what kind of a briefing he prepared for Trump ahead of the 25 July call.

Vindman says he based the talking points on official US policy about Russia, as well as anti-corruption and reform efforts.

"As you listened to the call did you observe whether President Trump was following the talking points based on the official US policy?" Goldman asks.

Vindman says Trump did not follow what he had provided.

Anthony Zurcher
BBC North America reporter

Image source, Getty Images

For Americans watching the impeachment hearings, it often seemed like the Democrats and Republicans weren’t just talking across each other, they’re not even existing in the same political universe.

That divergence of realities was on stark display from the start of Tuesday’s proceedings, as Democrat Adam Schiff and Republican Devin Nunes gave their opening remarks.

As he had in the previous two days of hearings, Schiff used his time to lay out what he views as the case against Donald Trump. He described the president’s alleged “scheme” to pressure the Ukrainian government to open investigations of Democrats that could be politically beneficial to him.

He noted witness testimony of instances of this Ukrainian pressure. He cited Saturday’s closed door deposition by US Ukraine-based diplomat David Holmes, who said he overheard a call during which Trump asked US Ambassador to EU Gordon Sondland about the "investigations". Sondland, Holmes said, would later observe that the president didn’t care about Ukraine beyond how the investigations could help him personally.

Nunes, in his five-minute statement, declined to offer a line-by-line defence of the president. Instead, he went on the attack.

He said media coverage of last week’s impeachment hearings were biased in favour of the Democrats. He called out CNN, the Guardian, Slate, the Daily Beast, New York magazine and others by name. He said their coverage of Robert Mueller’s special counsel investigation into Russian meddling in the 2016 presidential election was frequently wrong, and the Ukraine story was simply a continuation of a campaign to oust the president.

Democrats, in these hearings, are trying to set out a series of facts that they say point to presidential abuse of power and could constitute an impeachable offence. The Republican strategy, at least at this point, is to encourage Americans that they can’t trust the facts – or the mainstream media outlets that are reporting them.

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Williams says the reference in the 25 July call to Joe Biden in particular "struck me as political in nature".

Both Williams and Vindman said there was "no credible evidence" to support Trump's charges against Joe Biden and his son.

Joe Biden's involvement in Ukraine during his tenure as vice-president is a key part of the Trump-Ukraine affair.

While Joe Biden served as vice-president, his son joined the board of a Ukrainian energy company, Burisma. It raised eyebrows at the time because of perceived conflict of interest.

Trump and his supporters allege Biden abused his power to pressure Ukraine to back away from a criminal investigation into that firm in case such a probe would implicate his son, Hunter.

But there is no evidence there was an active investigation into the gas firm, nor any evidence Biden took any action to intentionally benefit his son.

Biden had bragged about removing the Ukrainian chief prosecutor, but he was not alone in calling for his removal - many lawmakers in the US and EU also wanted him ousted because of how little he was doing in tackling corruption.

Watch video below for more on this.

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Schiff asks both Williams and Vindman why the word "Burisma" - the Ukrainian gas company that employed Hunter Biden - was missing from the 25 July call transcript, when both witnesses recall that company being mentioned by name by President Zelensky.

"I attribute that to the fact that the transcript... may have not caught the word," Vindman says.

He says the word "Burisma" was represented as "the company" in notes of the Trump-Zelensky call.

"The folks that produce these transcripts do the best they can," Vindman says.

He continues: "It is not a significant omission."

Vindman says he did not regard the omission as "nefarious".

Image source, Chip Somodevilla/Getty Images

Williams and Vindman both say they were told American military support of Ukraine was suspended to ensure it remained consistent with the administration policies.

"It was not made more specific than that," Williams says.

Vindman continues that Ambassador Gordon Sondland - a Trump loyalist set to testify on Wednesday - told the Ukrainians that in order to get a White House meeting, the Ukrainians would have to provide a "deliverable, which is investigations".

Chairman Schiff asks Vindman if he gave Ukraine's President Zelensky any advice when he met him.

"I offered two pieces of advice: to be particularly cautious with regards to Russia and its desire to provoke Ukraine and the second one was to stay out of US domestic politics.

"In the March and April time frame it became clear that there were actors in the US, public actors, non-governmental actors, that were promoting the idea of investigations and 2016 Ukrainian interference and it was consistent with US policy to advise any country... to not participate in US domestic politics."

Fox & Friends - Trump's favourite morning programme - offered him this some timely advice this morning.

"The president should just ignore this whole thing," said host Brian Kilmeade.

"Don't tweet during it. Don't get outraged over it. It ticks you off.

"There's so much for him to do."

Image source, Win McNamee/Getty Images

Ending his statement on a personal note, Vindman addresses his father directly, thanking him for his "brave act of hope" to leave the Soviet Union and start over in the US with his three sons.

"Dad, my sitting here today, in the US Capitol talking to our elected officials is proof that you made the right decision forty years ago."

He continued: "Do not worry, I will be fine for telling the truth."

Vindman and his two brothers are all currently serving in the US military.

Vindman condemns "character attacks" against his colleagues who have appeared before lawmakers already or are scheduled to do so.

"I want to state that the vile character attacks on these distinguished and honourable public servants is reprehensible. It is natural to disagree and engage in spirited debate, this has been our custom since the time of our Founding Fathers, but we are better than callow and cowardly attacks."

He does not elaborate on these attacks, but during former ambassador Marie Yovanovitch's hearing, the president caused a stir by assailing her in a tweet.

Democrats accused the president of intimidating a witness, and even some Republicans felt the tweet had gone too far.

Read BBC North America reporter Anthony Zurcher's analysis of how that Trump tweet shook the hearing here.

Vindman says he heard the July call between Trump and Zelensky in the Situation Room along with other White House colleagues.

"I was concerned by the call, what I heard was improper, and I reported my concerns," he says.

"It is improper for the President of the United States to demand a foreign government investigate a US citizen and political opponent."

Vindman adds that if Ukraine had opened such an investigation, it would play into Russia's objectives.

"I want to emphasise to the committee that when I reported my concerns - on July 10, relating to Ambassador Sondland, and on July 25, relating to the President - I did so out of a sense of duty. I privately reported my concerns, in official channels, to the proper authorities in the chain of command."

He adds: "I never thought I would be sitting here testifying in front of this committee and the American public, about my actions."

Image source, Getty Images

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Vindman describes an early July visit with Ukraine's national security adviser, where US Ambassador to the EU Gordon Sondland made what he called "inappropriate" comments about the investigations into the Bidens and election meddling.

Vindman says they anticipated Ukraine would discuss a meeting between Trump and Zelensky.

But Sondland began to speak about "the requirement that Ukraine deliver specific investigations in order to secure the meeting with President Trump", which led former security council chief John Bolton to cut the meeting short.

"Following this meeting, there was a short debriefing during which Ambassador Sondland emphasised the importance of Ukraine delivering the investigations into the 2016 election, the Bidens, and Burisma. I stated to Ambassador Sondland that this was inappropriate and had nothing to do with national security."

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Lt Col Vindman is the principal adviser to the National Security Adviser and the US president on Ukraine.

He begins his opening statement by noting his background - including two decades of service in the US Army, and served in the US embassies in Kyiv, Ukraine, and Moscow, Russia.

He says he became aware this spring of "two disruptive actors - primarily Ukraine's then-prosecutor general Yuriy Lutsenko and former New York City Mayor Rudolph Giuliani, President Trump's personal attorney who he said were peddling "false narratives that undermined the United States' Ukraine policy".

Image source, Chip Somodevilla/Getty Images

In her opening statement, Jennifer Williams, an aide to Mike Pence and a career diplomat, says she has worked with the office of the vice-president for the last eight months.

She says: "I found the July 25th phone call unusual because, in contrast to other presidential calls I had observed, it involved discussion of what appeared to be a domestic political matter."

She says she provided details of the call to the vice-president's daily briefing book, but she does not know if Pence read her transcript.

Trump is at the White House all day where he has lunch with Vice-President Pence and a cabinet meeting.

The president has tweeted once this morning, to boast of the US stock market's performance.

But his top campaign spokesman has tweeted about this morning's hearing.

Tim Murtaugh argues that the controversy over the Trump-Zelensky call boils down to "policy disagreements".

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.

In his rebuttal to Adam Schiff's opening statement, ranking Republican Devin Nunes (a Trump defender) pilloried the "mainstream media", "puppets of the Democratic party" who "reported breathlessly on the new bombshell allegations".

The California congressman suggested "co-ordination" between Democrats and the whistleblower whose report sparked the impeachment inquiry.

No evidence has come to light that the anonymous intelligence official worked with Democrats.

Read BBC Reality Check's assessment on three key Nunes claims

Schiff also says both witnesses currently before lawmakers, Vindman and Williams, "took note of the explicit use of the word Burisma by Zelensky, a fact conspicuously left out of the record of the call, now locked away on a secure server".

Burisma is the Ukrainian energy company that Joe Biden's son, Hunter, worked for while his father was vice-president.

Trump is accused of wanting Zelensky to pursue an investigation of Burisma.

The word "Burisma" does not appear in a rough transcript of the July phone call released by the White House.

In his opening statement, Democratic intelligence committee chairman Adam Schiff - who is overseeing the impeachment inquiry - said: "How could our diplomats urge Ukraine to refrain from political investigations of its own citizens, if the president of the United States was urging Ukraine to engage in precisely the same kind of corrupt and political investigations of one of our own citizens?"

Schiff notes that Vindman testified earlier about how, due to Ukraine's dependency on the US, "the favour Trump asked of Zelensky was really a demand".