Elections are held for 248 English councils, six mayors and all 11 councils in Northern Ireland
Conservatives and Labour both see vote shares fall compared with 2015
The Tories lose 44 councils and 1,334 seats
Theresa May says results message is: "Just get on and deliver Brexit."
Labour loses six councils and 82 seats
Lib Dems take control of 10 councils and won 703 seats
Greens and independents also make significant gains
Anger over Brexit appears to be influencing voters, say experts
Professor Sir John Curtice
Polling expert
So far turnout is averaging just one to two points below what it was in 2018 and 2016, so one thing that the Brexit impasse has not done is to drive voters away from the polls in significant numbers.
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Professor Sir John Curtice
Polling expert
There are signs that the Lib Dems may be restoring their support, in particular in the kinds of places that would once have been regarded as the party's core constituency - that is, socially liberal university graduates who might have been upset by the party's volte face on tuition fees.
The party's vote is up on average compared with 2015 by as much as 11 points in wards where more than 30% of the population have a university degree, whereas it is up by only 3 points in places where less than 20% have a degree.
Conversely, the Conservatives' support seems to be falling most heavily in places with large numbers of university graduates whilst the party's vote has held up much better in places with few graduates.
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Professor Sir John Curtice
Polling expert
"A further indication of the apparent 'plague on all your houses' mood that seems to be expressed in these elections is the fact that independent candidates have so far secured an increase of 43 seats - representing a trebling of their representation in the same wards before today."
Tories lose their fifth council
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BBC News Channel
Labour MP Margaret Beckett is speaking from Derby, where she says it has been a "disappointing" night for her party.
She says the "wild card factor" is the voter identification trial that has been taking place in Derby.
"It is much too early to have clear evidence but some people are saying that has put them off from going to vote," she says.
Conservatives lose their fourth council
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The Liberal Democrat hold their first council
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The Conservatives lose a third council
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Follow all the results as they come in with our interactive map.
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Laura Kuenssberg
BBC political editor
So far, it is not the kind of night that Labour wanted to have. At this stage in the electoral cycle they should be making big strides.
Conservatives expect things to be grim. Their results are not awful, but not great.
It is not a good night for either of the big parties.
Both the Lib Dems and Greens are gaining votes.
The really interesting thing is that they are not just gaining in Remain areas.