* Opinion polls show support for "In" and "Out" almost equal

* Two show swing towards "Remain" late on Wednesday

* Political fallout will reach beyond Britain

* Financial volatility likely either way

By Elizabeth Piper

LONDON, June 23 (Reuters) - Britons will decide the future of their country and Europe on Thursday in a vote on European Union membership after a bitter campaign that appeared to divide the nation down the middle.

Prime Minister David Cameron called the referendum under pressure from his ruling Conservative Party and an increasingly powerful anti-EU party, hoping to put to rest decades of debate over Britain's place in Europe and its ties with Brussels.

Most opinion polls put the "Leave" and "Remain" camps neck-and-neck at the end of a campaign that was dominated by immigration and the economy and shaken by the murder of a pro-EU MP, though late on Wednesday two showed a swing to "Remain".

The "Leave" campaign says Britain's economy would benefit from a Brexit, or British exit. Cameron says it would cause financial chaos.

Traders, investors and companies are preparing for volatility on financial markets whatever the outcome of a vote that both reflects, and has fuelled, an anti-establishment mood also seen in the United States and elsewhere in Europe.

Much will depend on turnout, with younger Britons seen as more supportive of the European Union than their elders but less likely to vote.

"Go out and vote remain for a bigger, better Britain inside a reformed European Union," Cameron told "Remain" campaigners on Wednesday.

His main rival, former London mayor Boris Johnson, whose decision to support "Leave" galvanised its campaign, told voters this was the "last chance to sort this out".

Sterling rose to its highest so far this year against the U.S. dollar late on Wednesday after one poll pointed to a clear lead for "Remain" and betting markets priced in an 80 percent chance Britain would not leave.

Polling stations for 382 local counting areas will open at 0600 GMT and close at 2100, with most of the results expected between around 0100 and 0300 on June 24.

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 how 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.

"If we don't vote to leave tomorrow we will remain locked in the back of the car, driven in an uncertain direction, frankly, to a place we don't want to go and perhaps by a driver who doesn't speak the very best of English," said Johnson, a leading candidate to replace Cameron as prime minister.