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Liberal Democrat leader Sir Ed Davey has hailed the local election results as a "tuning point" for his party's prospects.
With over half of councils in England, Wales and Scotland declared, the Lib Dems have over 100 new councillors, the biggest net gain of any party so far.
They have made gains in Tory heartlands in southern England, as well as seizing Hull City Council from Labour.
Remaining council results across Britain are expected by Friday evening.
Results are also due on Friday and Saturday from Northern Ireland, where voters have been electing members of the Northern Irish Assembly.
Labour leader Sir Keir Starmer has said his party is "back on track" for the next general election after taking key London councils from the Tories.
Boris Johnson has said the Conservatives had had a "tough night" in some areas, but argued results overall had been "mixed".
The Lib Dems have gained seats in Wokingham in Berkshire, where it has tipped the previously Tory-majority council into no overall control.
The same in true in West Oxfordshire, whilst the Lib Dems have also displaced the Tories as the largest party in Tunbridge Wells in Kent. They have also taken overall control in Somerset, a new single-tier authority that had been a big target for the party.
The party has also taken back control of Hull City Council from Labour, where they had been fighting to regain power since 2011.
Polling expert Sir John Curtice said the Lib Dems' gains reflected its relatively poor performance in 2018, when most of the seats up for grabs this time were last contested.
He added that the Lib Dems' gains reflected the decline in Conservative support since then, with its advance most prominent in areas where it was in second place to the Tories.
The BBC projects the Lib Dems would take 19% of the votes had all parts of Britain gone to the polls - matching its best local election performance since it entered coalition government with the Conservatives in 2010.
The Liberal Democrats had entered the local elections hoping to build on by-election successes over the Tories in Chesham and Amersham and North Shropshire last year.
Speaking to party activists in Wimbledon, south-west London, Sir Ed said the victories for his party were an "almighty shockwave that will bring this Conservative government tumbling down".
He added that voters had "had enough" of the squeeze on living standards, along with the government's decision to raise National Insurance from April.
He also said voters had responded positively to his party's plans to cut VAT using the proceeds from a windfall tax on the profits of oil and gar companies.
Labour, which has also backed a windfall tax on energy companies, had also put the cost of living at the centre of its campaign.
'Tough night'
It has gained symbolically important London councils from the Conservatives such as Westminster and Wandsworth, but its fortunes have been more modest outside the capital.
Speaking to reporters earlier, Boris Johnson admitted the Conservatives had endured a "tough night in some parts of the country".
But, arguing the results overall were "mixed", he added: "In other parts of the country you are still seeing Conservatives going forward and making quite remarkable gains in places that haven't voted Conservative for a long time, if ever."