The Remain side holds a narrow lead in the final poll of the knife-edge EU referendum campaign.
The In camp are ahead of Leave by 52 per cent to 48 per cent, according to an Ipsos Mori survey for the Evening Standard.
To add further uncertainty to the outcome, 12 per cent of those polled said they could switch sides as they head for the polling booth.
With turnout key to the result, torrential rain storms in Remain stronghold London could deter voters casting their ballots.
There were long queues snaking outside voting centres as the nation braved the storms to decide Britain's future role in Europe. But some polling stations were forced to close, and many others were flooded due to the heavy rain sweeping the capital. Kingston upon Thames Council in southwest London moved two polling stations after they became inundated with water.
Prime minister David Cameron ignored questions about the weather, saying only "Good morning" to the gathered media from across the world, as he and wife Samantha cast their votes at Methodist Hall in Westminster.
Labour leader Jeremy Corbyn arrived in a positive mood to cast his vote at Pakeman Primary School in his Islington constituency in north London.
Asked if he was feeling confident, Mr Corbyn smiled and said: “Extremely, it’s a very good day.”
On the outcome of the referendum, he joked: “You could either check the wind or check the bookies,” adding“the bookies usually get it right”.
Mr Corbyn acknowledged that the odds on his election as party leader last year had not been favourable, grinning as he said: “I cost them [bookmakers] a lot of money.”
‘Soft Remainers’
The Leave camp has a "very strong chance" of pulling off one of the biggest political upheavals of recent times, Ukip leader Nigel Farage insisted.
“I do think we are in with a very strong chance, I do genuinely,” Mr Farage said outside his Kent home. “But it’s all about turnout and those soft Remainers staying at home.”
The referendum has seen one of the longest, and most personally bitter, campaigns in recent British political memory.
Both sides of the campaign have been fighting for months, and things came to a frenetic close on Wednesday as senior politicians criss-crossed across the country to try to sway undecided voters.
Mr Cameron and his Remain colleagues from across the political spectrum have warned of the potentially severe economic consequences of a vote to leave the EU amid fears of financial market turmoil and another recession.
But Leave campaigners, led by Conservative former mayor of London Boris Johnson, have urged voters to "take back control" of the country.
They believe a divorce from Brussels would give the UK more money to spend on national issues such as funding the NHS as well as giving the government the ability to control the nation's borders and levels of immigration.
The campaign has been punctuated by ill-tempered exchanges and interventions, with both sides accusing the other of scaremongering, particularly over the issues of the economy and immigration.
‘Breaking point’
Mr Farage would not elaborate on his reasons for missing Wednesday night’s final Channel 4 debate of the campaign, saying only that it was “for family reasons”.
He came in for particularly stern criticism after unveiling a Brexit poster showing a queue of hundreds of immigrants arriving in Europe with the slogan "breaking point".
Leave campaigners were left furious after chancellor George Osborne made use of treasury research to warn that quitting the EU would result in households being £4,300 (€5,600) a year worse off.
Meanwhile, dozens of celebrities have intervened during the course of the campaign to make their feelings known.
Former footballer David Beckham, James Bond actor Daniel Craig and Harry Potter author JK Rowling were just three of the high-profile names to back the Remain campaign, while Leave won support from the likes of comedian John Cleese, former cricketer Ian Botham and former footballer Sol Campbell.
The polls in the run-up to the vote have suggested the result of only the third nationwide referendum in UK history will be tight, with neither side able to surge ahead in the final weeks.
A record number of voters are eligible to take part in the referendum, with the Electoral Commission putting the number at 46,499,537.
Northern Ireland
Stormont’s First Minister has expressed hope of a high turn-out in Northern Ireland as voters continued to cast their ballots in the EU referendum.
Arlene Foster, a Brexit campaigner, said it was important that those on both sides of the debate made their voices heard.
“I understand there has been a good turn-out in some of the areas where previously there hasn’t been so far - so that’s good to hear,” she said.
The Democratic Unionist leader added: "Everyone has the opportunity to cast their vote today and I hope they take that opportunity, regardless of what that might be, because this is a huge opportunity for all the citizens of the United Kingdom to make their voices heard in terms of the European Union."
Sinn Féin president Gerry Adams, a Remain advocate, also encouraged people to get out and vote.
“Come out and vote, if you haven’t voted there is still time, vote to remain - it’s not in Ireland’s interests to do anything else,” he said.
Polling stations opened at 7am and will close at 10pm.