The Brexit Party was the clear winner in the UK's European elections, with the pro-EU Lib Dems coming second.
The Conservatives and Labour suffered heavy losses, with the former expected to get less than 10% of the vote.
Brexit Party leader Nigel Farage said he was ready to "take on" the Tories and Labour in a general election.
Overall, out of 64 MEPs declared so far, Mr Farage's party has won 28, the Lib Dems 15, Labour 10, Greens seven, the Tories three and Plaid Cymru one.
Mr Farage told BBC Radio 4's Today programme: "With a big, simple message - which is we've been badly let down by two parties who have broken their promises - we have topped the poll in a fairly dramatic style.
"The two party system now serves nothing but itself. I think they are an obstruction to the modernising of politics... and we are going to take them on."
The deputy leader of the pro-Remain Liberal Democrats, Jo Swinson, said the result should "give hope to all of the people out of there who want to stop Brexit", and that they "should not be giving up without a fight".
"When there is such a general shambles over Brexit, then [a further referendum] is the best way out of the impasse," she told Today.
Polling expert Sir John Curtice said the results showed just how polarised the country had become.
All 28 EU member states are electing MEPs to the European Parliament - the EU's law-making body. Overall, the big centre-right and centre-left blocs lost ground, amid a surge in support for liberals, Greens and nationalists.
The UK had been due to leave the EU on 29 March, but when that deadline was missed, participation in the election became mandatory.
The Brexit Party topped the polls in every region of England apart from London. It also dominated in Wales, with Plaid Cymru second.
It has now become the joint largest national party in the European Parliament, alongside Germany's CDU/CSU party.
Scotland is yet to formally declare, but with most votes counted, the SNP is on course for a resounding victory, with about 38% of the vote.
Meanwhile, Northern Ireland's count does not begin until Monday, with the results expected on Tuesday.
The night at a glance
- The UK is electing 73 MEPs, and 10 out of 12 regions have declared so far
- The Brexit Party has received the highest share of the vote in nine of the 10 regions - with 32% overall
- The Conservative Party has been widely rejected by the electorate, coming in fifth place
- The Lib Dems have taken second place with 20% of the vote - a vote that is clearly much higher in areas that backed Remain. In London, the Lib Dems came first and the Tories fifth
- The Labour Party has fallen to third place overall - fifth in Scotland - and is on course to end up with less than 15% of the vote, an even worse performance than their previous low in 2009
- The Green Party has also made gains, seeing its best performance since 1989
- Newly-formed Change UK has not won any seats, while UKIP has also haemorrhaged support to The Brexit Party - both have gained about 3% of the vote
The Tory story
BBC political editor Laura Kuenssberg said it was the worst performance for the Conservatives since the 1830s.
"If this was a first-past-the-post election, they would not have taken a single seat," she said.
Brexiteer and Conservative MEP for the South East Daniel Hannan - one of only three Tory MEPs elected so far - agreed it was his party's "worst ever result".
"We voted to leave (the EU) and we haven't left - it's that simple," he told the BBC.
Theresa May tried three times to secure MPs' backing for her Brexit plan and resigned as prime minister after her fourth attempt prompted a big backlash.
Tory leadership hopeful and former Foreign Secretary Boris Johnson said the public had delivered "a crushing rebuke" to both major parties for failing to deliver Brexit.
Prominent Brexiteer and Tory backbencher Steve Baker said his party faced "obliteration" if they did not get on with leaving the EU as they had pledged.
"We have lost some excellent MEPs who didn't deserve to lose," he told BBC Breakfast.
"It is a grave time for our party and country, [but] I am not surprised that [the result] is as bad as it is. You just can't break promises you have made like this and expect to go unpunished."
Labour leader Jeremy Corbyn placed blame for that failure squarely with the Conservatives, arguing it meant the elections had "become a proxy second referendum".
He said his party would "reflect on these results on both sides of the Brexit divide" and the issue "will have to go back to the people" through a general election or a public vote.
Labour's deputy leader Tom Watson said the party needed to "urgently" rethink its Brexit position after the "disastrous" election results.
The party has been criticised for a lack of clarity on its Brexit stance and not appearing to appeal to either Leave or Remain voters. It has been divided over the issue of a further referendum, with some - including shadow foreign secretary Emily Thornberry - saying the party should call for another vote and campaign to remain.
Tweeting on Monday morning, shadow chancellor John McDonnell said Labour "must unite our party and country by taking issue back to people in a public vote."
Alastair Campbell, former spin doctor to Tony Blair and prominent member of the People's Vote campaign for another referendum, said for the first time in his life he had voted for the Lib Dems.
Big gains for smaller parties
In contrast to the two biggest parties, some of the smaller ones with an unambiguous message on Brexit were rewarded.
Anti-Brexit parties - those in favour of another referendum - collectively took about 40% of the vote, compared with 35% for the two parties in favour of leaving the EU without a deal.