We’ve heard a lot in recent weeks about Russia’s alleged involvement in the EU referendum.

On Monday, Theresa May said Vladimir Putin’s government was trying to “undermine free societies” and was “planting fake stories” to “sow discord in the West”.

Although on Wednesday, the Prime Minister appeared to soften her language. She told parliament: “if [Labour MPs] cared to look at the speech I gave on Monday they will see that the examples I gave of Russian interference were not in the United Kingdom.”

Russia has been accused of meddling in recent elections in America, France and elsewhere. What about Britain?

FactCheck takes a look at what we know.

Alleged Russian influence on US politics through social media

To understand the allegations of Russian involvement in the EU referendum, we should first look at its reported links to the US presidential election.

Many of the allegations centre on the Russian Internet Research Agency. The company, based in Saint Petersburg, creates online media posts that are favourable to Russian government interests.

The Agency was making the news as far back as April 2015, when Donald Trump was still “the guy off the Apprentice” and “Brexit” was a word muttered only in the back rooms of the Conservative Party headquarters.

At the time, the Guardian published an investigation “Inside a Russian troll house”. They found “hundreds of bloggers [who] are paid to flood forums and social networks at home and abroad with anti-western and pro-Kremlin comments”.

It is believed that this is the home of the Agency and the nexus of Russia’s attempts to use social media to influence overseas politics.

Jump forward to January 2017 – six months post-referendum and six weeks after Mr Trump’s election. The Office of the US Director of National Intelligence publishes a report that concludes “Russian President Vladimir Putin ordered an influence campaign in 2016 aimed at the US presidential election”.

In October this year, we found out more about how the Agency works, after a former employee told an independent TV channel that the company’s goal “wasn’t to turn Americans toward Russia. Our goal was to set Americans against their own government… to provoke unrest, provoke dissatisfaction, lower [Barack] Obama’s rating”.

And that month, we got an insight into the scale of Russia’s alleged social media influence.

The New York Times reported that “Russian agents … disseminated inflammatory posts that reached 126 million users on Facebook, published more than 131,000 messages on Twitter and uploaded over 1,000 videos to Google’s YouTube service” ahead of the US presidential election.

The reports were drawn from “detailed disclosures sent to Congress … by companies whose products are among the most widely used on the internet”.

Earlier this month, a BuzzFeed investigation found that the Russian government made wire transfers to more than 60 Russian embassies between August and September 2016. BuzzFeed reported that the transfers were sent with the message “to finance election campaign of 2016”.

A spokesperson for the Russian government said the money was designed to help embassies cover the cost of helping Russian citizens living abroad to vote in the country’s own parliamentary election, which took place in September 2016.

What is not clear is whether any of this activity was linked to Donald Trump’s presidential campaign – that question will be answered in the course of the five federal investigations that are underway in the US.

It’s also worth pointing out that some attempts to uncover Russian-controlled fake accounts have been inaccurate. A report by the Scotsman found that “an alleged Russian online troll named on a prominent news website is actually a security guard from Glasgow”.

Did Russian fake accounts try to influence the EU referendum?

So, according to reports, the Russian Internet Research Agency built up a significant social media presence in the run up to the US presidential election. It was apparently able to reach hundreds of millions of users.

But what about the UK? What evidence do we have so far of Russian influence on the Brexit vote?

In October 2017, research from City, University of London found that a “13,500-strong Twitter bot army” was present on the social media site around the time of the referendum.

Twitter bots are accounts that are not run by real people, but instead post content automatically. The accounts sometimes adopt a fake persona to attract real users to their posts. You can find out more about how to spot a bot here.

The lead researcher of the City project said of the bots: “We believe these accounts formed a network of zombie agents.” He said “We didn’t find evidence that bots helped spread fake news. Instead, they were invested in feeding and echoing user-curated, hyper partisan and polarising information”.

In the four weeks prior to the vote, the accounts posted a total of 65,000 tweets about the referendum – tweets that were described as showing a “clear slant towards the Leave campaign”.

However, it’s worth saying that the researchers didn’t mention any Russian involvement.

This week, further research into Twitter bots and the Brexit vote was published by the Oxford Internet Institute. It looked at 22.6 million tweets and cross-referred them with 2,752 accounts that the US Senate has identified as creations of the Russian Internet Research Agency.

Researcher Yin Yin Lu told Sky News that she had found 416 tweets from the Russian accounts from March to July 2016 (i.e. the months preceding the EU referendum).

She was careful to point out that “the number of these tweets is important to highlight. So there’s about 400 tweets here out of 22.6 million. That is a very infinitesimal fraction. So the word ‘interference’ is perhaps a bit exaggerated”.

Although she did say that the tweets did appear to be coordinated – with accounts apparently retweeting and sharing each other’s content. She said “there’s some kind of network happening here”.

A second report – this time from the University of Edinburgh – found a higher number of tweets about Brexit from the Russian Internet Research Agency. The Guardian reports that researchers identified 419 accounts operating from the Agency that were attempting to influence UK politics. The accounts were on the list of 2,752 accounts suspended by Twitter in the US.

Professor Laura Cram, who led the research, told the Guardian that those 419 accounts tweeted about Brexit at total of 3,468 times – mostly after the referendum had taken place.

Separately, an upcoming paper from researchers at the University of California, Berkeley, and Swansea University is set to reveal an even wider network of pro-Brexit Russian bots.

The research – which has been seen by the Times newspaper – tracked over 150,000 Russian accounts that used the hashtag “Brexit”, most of which were advocating Britain’s departure from the EU.

One account, “Svetal1972”, posted 92 tweets between 20 and 24 June, including one that called for Britain to “make June the 23rd our Independence Day”.

But why would Russia want Brexit to happen anyway?

So there’s some evidence of Russian involvement in spreading pro-Leave sentiment online – although nothing as yet that suggests interference on the same scale as is alleged in the US. But if those reports are true, what’s in it for the Kremlin?

Some analysts have suggested that Russia favours Brexit because it thinks it will weaken the EU’s negotiating position when it fronts up to Moscow on issues like the Ukraine.

This theory seems plausible when we look at the tweet from the mayor of Moscow, Sergei Sobyahin, who wrote that “Without Britain, there won’t be anybody in the EU to defend sanctions against us so zealously”.

Michael McFaul – a former US ambassador to Russia – put the case in similar terms, tweeting “Putin benefits from a weaker Europe. UK vote [to leave the EU] makes EU weaker. It’s just that simple”.

What about Arron Banks?

Separate to the question of social media influence, Arron Banks – the co-founder of the Leave.EU campaign – is currently under investigation by the Electoral Commission.

They are looking into whether Banks broke campaign finance rules in relation to donations and loans at the EU referendum.

The company Better for the Country Limited – of which Banks is a director – is also part of the probe.

The investigation is looking at whether Banks was the “true source of loans reported by a referendum campaigner in his name” and whether “Better for the Country Limited was the true source of donations made to referendum campaigners in its name, or if it was acting as an agent”.

The announcement by the Electoral Commission from November this year makes no mention of Russia or any Russian actors. Indeed, it doesn’t say whether there are any specific third parties being examined as part of the investigation.

In a statement, Mr Banks said: “The Leave. EU campaign was funded by myself, Peter Hargreaves and the general public . . . My sole involvement with ‘the Russians’ was a boozy 6 hour lunch with the Ambassador where we drank the place dry.”

In an interview with Banks earlier this year, the Guardian quoted him as saying that “we had no Russian money into Brexit”.

The journalist conducting the interview, Carole Cadwalladr, wrote that the comment “would be a perfectly reasonable answer, if he had been asked if Russia had put money into Brexit. But he hadn’t. He asked and answered his own question”.