Labour leader Jeremy Corbyn says he is "very sorry" to have lost 80 councillors, but adds: "We'll fight back and we'll win them back."

He says the results so far show Conservatives have lost about 500 councillors and swings to Labour in places such as Trafford, where he was visiting, give the party a platform for future elections.

"We are going to campaign and continue to campaign to try and keep and bring people together in negotations over Brexit, but also to oppose austerity and put forward an agenda which deals with the injustices that exist in this country," he says.

He acknowledges that the party want to do better, and blamed the results on "local factors" as well as "people probably disagreeing with both parties on attitudes to the European Union".

"Our policy is that we’re the only party that seeks to appeal to people however they voted in 2016 and to seek to defend jobs and working conditions in this country," Mr Corbyn says.

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Green Party co-leader Jonathan Bartley says it has been "the biggest election night in our history".

The party has gained 42 councillors so far.

"Voters see that we need a new kind of politics, one that recognises the huge imperative of acting on climate change, but also the social emergency that is creating misery and suffering in communities across the country," Mr Bartley says.

He adds that Brexit has also played "a significant role" in the elections, with the Greens "standing up for the UK's membership of the European Union".

Although Labour held on to Sunderland Council, the party lost 12 seats.

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Elections analyst Prof John Curtice says that both Conservatives and Labour were "punished" by voters in local elections, with a similar drop in the share of the vote.

"We knew already that the Conservatives were in deep trouble as a result of their failure to deliver Brexit, but it also looks as if the Labour Party is being punished, because of the dissatisfaction with the way it's been reacting to the Brexit impasse," he tells BBC News.

But, he says, parties suffered most "where they were seen as part of the establishment", with Labour's vote falling most in the north and the Conservatives in the south.

He says it is a "wider protest" than Leave supporters abandoning Conservatives and Labour.

Meanwhile, the Liberal Democrats made gains in places where they had previously been strong, not in the most pro-European areas, Prof Curtice says.

He says they were "beginning to act once again as the party of protest, which used to be their traditional role until they went into coalition with the Conservatives".

UKIP's vote was down from its peak in 2015, but higher than last year, when the party "came close to disappearing".

"We do have evidence that the Eurosceptic vote is back up," he says. "There’s a warning sign for the Conservatives."

Labour Party chairman Ian Lavery says that anger over austerity was "trumped" in these local elections by anger over Brexit.

"We’ve got a clear message from those communities that the two parties need to get on and get Brexit sorted," he tells BBC News.

"There’s been huge anger about Tory austerity, but there’s more anger that this government hasn’t been able to deliver the Brexit that they promised."

He says people in the Midlands and further north "voted for change, they voted for smaller parties" in a "protest vote". "I share their frustrations," he says.

"The reality about Brexit is it’s been extremely divisive and the Labour Party are the only party who are trying to bring both sides of that divide together. It’s essential we bring this country together."

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Conservative Party deputy chairwoman Helen Whately says she does not accept that the local elections are a negative verdict on Theresa May's leadership.

"On the doorstep I heard people who were frustrated with us that we hadn’t delivered Brexit, it wasn’t a frustration with the prime minister," she says.

"It was overall a frustration with Parliament for not seeing through the referendum result."

She said the strongest message for MPs was "we've got to get on and do Brexit", but because of the divisions in the country "we have to do a moderate Brexit that works for everyone".

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Ellie, who canvassed for the local elections for the Conservatives in north Somerset, says she is "not surprised" by her party's results.

“The impression on the doorstep from voters who had always been Conservative was they said they wouldn’t vote for us because of how things are going in Westminster," she tells BBC Radio 5 Live.

She adds that people told her: "I’m not voting for you until you get us out the EU and have delivered Brexit."

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Conservative MP for Witham Priti Patel says that the party should consider a change of leadership.

She tells BBC News that this was now the "ultimate question" for the Conservatives and voters on the doorstep have made their views clear about Theresa May.

"People have very categorically said, she is part of the problem," she says.

"I think we need change, I just don’t think we can continue like this," she says. "Many of my constituents have said this to me, we need change, we need a change of leadership. Perhaps the time has now come for that."