Prior to this week's polls, in the period between June 10 and June 16, "Out" led in seven of nine polls, rattling markets around the world.
Betting odds, provided by gambling firm Betfair on Thursday, gave the "Remain" campaign an 86 percent probability of winning, up from as low as 59 percent last week.
Britons were voting on Thursday on whether to stay in the European Union in a referendum that has split the nation and is being nervously watched by financial markets and politicians across the world.
An Ipsos MORI poll for the Evening Standard newspaper, conducted on Tuesday and Wednesday, suggested support for "Remain" stood at 52 percent while "Leave" was on 48 percent.
Two other polls published late on Wednesday also suggested a modest late swing towards "Remain", but the overall picture was of a vote that was too close to forecast.
Much will depend on turnout, with younger Britons seen as more supportive of the EU than their elders but less likely to vote.
Polling stations opened at 0600 GMT and will close at 2100, with results expected to be announced by the 382 individual local counting areas between around 0100 and 0300 on Friday.
Prime Minister David Cameron called the vote under pressure from the anti-EU wing of his Conservative Party and the surging UK Independence Party (UKIP), hoping to end decades of debate over Britain's ties with Europe.
The "Leave" campaign says Britain's economy would benefit from an exit from the EU, or Brexit. Cameron says it would cause financial chaos and impoverish the nation.
He voted early, and said on Twitter: "Vote Remain - so that our children and grandchildren have a brighter future."
His main rival, former London mayor Boris Johnson, whose decision to support "Leave" galvanised its campaign, told voters on Wednesday this was the "last chance to sort this out".
It is only the third referendum in British history. The first, also about membership of what was then called the European Economic Community, was in 1975.
The four-month campaign, which has exposed bitter divisions in the ruling Conservative Party, was dominated by immigration and the economy, and shaken by the murder of pro-EU Labour lawmaker Jo Cox last week.
Sterling up
Traders, investors and companies were braced for volatility on financial markets whatever the outcome of a vote that both reflected, and has fuelled, an anti-establishment mood also seen in the United States and elsewhere in Europe.
Sterling gained 1.5 percent against the dollar on Thursday, breaking above $1.49 for the first time since December 2015 after the Ipsos MORI poll was released.
The likelihood of a "Remain" vote implied by Betfair betting odds stood at 86 percent, the firm said shortly after the poll.
Finance leaders from Group of Seven leading economies will issue a statement stressing their readiness to take all necessary steps to calm markets if Britain votes to leave, government officials with direct knowledge of the matter said.
Britain's AAA credit rating could be swiftly downgraded by Standard and Poor's if the Brexit camp prevails, S&P chief sovereign ratings officer Moritz Kraemer told German daily newspaper Bild.
On Wednesday, campaigners from both sides tried to win over the estimated 10 percent of the 46.5 million electorate who polls suggest had still not decided which way to vote.
The "In" campaign took aim at their rivals by saying a Brexit would hurt the economy, security and the country's status. The "Out" campaign said high levels of immigration could not be controlled inside the EU and it was time to bring powers back from Brussels to London.
Split nation
The killing of Cox, a 41-year-old mother of two young children, in her electoral district in northern England prompted a pause in the campaign and soul-searching about its tone. Cox's husband said she had been concerned about the coarsening of political dialogue.
The man charged with her murder, asked to confirm his name in a London court, responded: "My name is death to traitors, freedom for Britain".
Britain is deeply divided on EU membership, with big differences between older and younger voters, and between pro-EU London and Scotland and eurosceptic middle England.
That split was reflected in British newspapers' front pages. "Independence Day" was the front page headline of the Sun tabloid, Britain's biggest-selling newspaper, while the Daily Mirror warned "Don't take a leap into the dark".
The issue also dominated news bulletins far beyond Britain. In China, the Global Times, published by the ruling Communist Party's official People's Daily, warned Britain would lose its influence globally if voters backed Brexit.
Whatever the outcome of the vote, the focus on immigration to Britain, which has increased significantly in recent years, could worsen frictions in a country where the gap between rich and poor has also been widening.
If Britons choose to leave, Scottish leader Nicola Sturgeon has suggested Scotland may call a referendum on leaving the United Kingdom.
Even with a vote to stay, Cameron could struggle to repair the rifts in his party and hold on to his job.
Foreign leaders, from U.S. President Barack Obama to Chinese leader Xi Jinping, have called on Britain to remain in the EU, a message supported by global financial organisations, many company bosses and central bankers.
International banks have warned that the value of the pound could fall dramatically if Britain votes to leave and traders expect markets to be more volatile than at any time since the 2008-09 financial crisis.
The "Out" campaign says a fall in the value of the pound would boost exports and has found support among some financial specialists and small businesses. It has urged voters to ignore what it calls the "establishment" which it says has the most to lose from Brexit.
The EU has struggled with migration and economic crises and a Brexit vote would boost opposition to it within other member states.
"Stay with us," European Council President Donald Tusk told British voters on Monday.
"Without you, not only Europe, but the whole Western community will become weaker. Together, we will be able to cope with increasingly difficult challenges of the future."
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