Jeremy Corbyn has said Labour achieved "solid" English local election results despite not making big gains.
The party failed to take several targets from the Tories, including Wandsworth, but won back Plymouth and became the largest party in Trafford.
Theresa May praised Tory councillors after winning Barnet, Basildon and Peterborough. The party gained from a collapse in UKIP's vote.
The BBC's projected national vote share puts both parties on 35%.
Polling expert Professor Sir John Curtice told the BBC it was "a draw", so far as both parties' national performance was concerned. The Lib Dems are on 16% according the BBC estimate, which uses the results of key wards to estimate what a Britain-wide vote would have been.
- The final council to be declared was Tower Hamlets, which was gained by Labour from no overall control.
- Across the 150 council elections, Labour won 2,350 seats, up 77, the Conservatives won 1,332, down 33, the Lib Dems won 536, up 75, the Greens won 39, up eight and UKIP won three, down 123.
- The BBC estimate, based on 800 key wards, puts turnout at 36%, the same as in these wards four years ago.
- The Conservatives gained the London council of Barnet from no overall control, and held their flagship London councils of Westminster and Wandsworth. They also won Basildon and Peterborough, which had both also been under no overall control. But they have lost control of their northern flagship Trafford Council after 14 years of Conservative rule.
- Labour had high hopes for the night, and have won Plymouth from the Tories and gained Kirklees from no overall control but lost Nuneaton and Bedworth and Derby.
- The Lib Dems have gained four councils, winning neighbouring Kingston-upon-Thames and Richmond-upon-Thames in Greater London and South Cambridgeshire from the Tories and Three Rivers from no overall control.
- The Greens have gained seats - taking some from Labour in Sheffield and from the Tories in Richmond upon Thames.
- UKIP, which performed strongly last time most of the seats were fought in 2014, has seen its vote collapse. It had just three councillors elected, after losing 123.
- Sinn Féin has won the Westminster by-election in West Tyrone .
- Labour MP Dan Jarvis has been elected mayor of the Sheffield City Region.
What are the Conservatives saying?
The Conservatives held on to their flagship London councils, gained control of Redditch from Labour and Barnet, Basildon and Peterborough, which had been under no overall control. But they have lost control of Trafford in Greater Manchester, a key northern stronghold, where Labour is now the largest party, Plymouth, and Mole Valley, Surrey - as well as losing two London councils to the Lib Dems.
According to the BBC's projected national vote share, the party is three points down on what it achieved in 2017's county council elections but, after eight years in government, it is better than its performance in any of the local elections held between 2012 and 2014 and in 2016.
Prime Minister Theresa May visited Wandsworth on Friday morning - one of the Conservative's key London councils which the party held, despite losing eight seats.
She said: "Labour thought they could take control, this was one of their top targets and they threw everything at it, but they failed."
She said the party "won't take anything for granted" and would "build on this success for the future".
Party chairman Brandon Lewis acknowledged some disappointments, but told BBC Breakfast that voters had chosen the Tories after seeing "good, clear and strong leadership" from the prime minister.
Labour did pick up a seat in the Tory stronghold of Kensington and Chelsea Council, but the Conservatives retained control, despite criticism of their handling of the Grenfell Tower disaster. Tory leader of the council, Elizabeth Campbell, said they needed to "rebuild trust".
How have Labour reacted to the results?
Labour was defending the most seats - more than 2,200 - and had talked up its prospects in London, where it made gains in last year's general election.
The party gained control of Plymouth from the Conservatives as well as Kirklees, but lost control of two Midlands councils - Derby and Nuneaton and Bedworth - to no overall control. Its hopes of seizing control of Wandsworth and Westminster proved unfounded while the party lost ground in Hillingdon.
Labour's projected national vote share - at 35% - is its best such performance since 2012 and puts it neck and neck with the Conservatives - but in 2013, 2014 and 2016 it was estimated to be narrowly ahead of the Tories.
Leader Jeremy Corbyn said his party's win in Plymouth was a "fantastic result" and "a sign that Labour is back in this part of Britain".
Of the party's performance more widely, he said "obviously I'm disappointed at any places where we lost a bit of ground" but described it as a "solid set of results" which left the party "well placed to fight and win the next general election".
Deputy leader Tom Watson said Labour now had more councillors in London than at any time since the 1970s.
Shadow Local Government Secretary Andrew Gwynne said he believed the anti-Semitism row had had an impact on the result in Barnet, north London, which the Tories took back from no overall control.
Backbencher Jess Phillips said the party had to look at what had gone wrong in white working class areas in northern England while Chuka Umunna called for an election "post-mortem" to look at why expected gains hadn't materialised. Mr Umunna said the government was "divided and incompetent" yet Labour had not seen "the big win in Wandsworth or Kensington that was expected".
What about the Lib Dems?
The Lib Dems have won back control of Kingston-upon-Thames, in south-west London, which they lost to the Tories four years ago, after taking neighbouring Richmond - both areas had a large majority in favour of remaining in the EU. The party has also gained control of Three Rivers, south-west Hertfordshire, which had been under no overall control and has won South Cambridgeshire from the Tories.
The party also withstood a strong Conservative challenge to hold onto Sutton in south-west London - while it picked up nine seats in Kingston-upon-Hull.
Professor Sir John Curtice said the party's 16% projected national vote share was an improvement on its performances between 2013 and 2015 - but well down on the 25% or so it would achieve before it entered the coalition government in 2010.
Party leader Sir Vince Cable hailed the result in Richmond: "It's certainly the beginning of the comeback of the Lib Dems. It's not going to happen overnight - Rome wasn't built in a day.
"Morale is now very high. We have demonstrated that we can win when we are focused and we can only build on that."
Lib Dems would not have enough seats to take either Tories or Labour into coalition government in the House of Commons prediction based on #elections2018— BBC Daily Politics and Sunday Politics (@daily_politics) May 4, 2018
But one option could see 283 Labour and 43 SNP MPs just reach the 326 needed for a majority, explains Prof John Curtice pic.twitter.com/19WDiNQTyl
Good result for the Greens?
The Greens have made overall gains, including taking two seats off Labour in Sheffield, where the Labour-run council has been engulfed in a row over the felling of thousands of trees. They also took seats from the Conservatives in the London borough of Richmond-upon-Thames and added two new councillors in Trafford and one in Worcester.
Greens co-leader Caroline Lucas told the BBC: "I think the Greens have had a really good night and... even with a fraction of the resources, a fraction of the media time when people see Greens on the ground they like what they see and they elect more Greens. And that's what we are seeing from Richmond to Greater Manchester to the Midlands. We've had some great results."
But she acknowledged she was "incredibly disappointed" to lose five of the Greens' 10 seats on Norwich Council, blaming a "bit of a Corbyn surge" among younger voters.
UKIP vote collapses
The Eurosceptic party has, as predicted, haemorrhaged support, losing nearly all of the council seats it won in 2014, when it broke through into local government for the first time.
It put up candidates in only 540 seats compared with 2,193 in 2014, lost all ten seats it contested in Basildon, five in Thurrock and seven in Dudley. So far they have lost more than 100 seats and won two - including unseating Labour's council leader in Derby.
Former deputy chair Suzanne Evans said the fact it had taken two seats in Derby was a sign it could still "put the cat among the pigeons". Meanwhile its general secretary, Paul Oakley, compared UKIP to the Black Death and said the party "not all over at all".
"Think of the Black Death in the Middle Ages. It comes along and it causes disruption and then it goes dormant, and that's exactly what we are going to do. Our time isn't finished because Brexit is being betrayed."
What do the experts say?
By BBC Political Editor Laura Kuenssberg
They seem stuck with each other - so close, in fact, that our projected national share of the vote from the results would mean that, in a hypothetical general election tomorrow, the two main Westminster parties would receive exactly the same levels of public support.
For reasons we've discussed at length overnight and in recent days, we have to be careful about directly transposing what happened on Thursday into a theoretical nationwide election.
But while in one sense not very much happened in the local election - not very many councils and seats changed hands and neither of the main parties have (with a couple of hours to go) made dramatic strides - something important did happen.
Voters opted again for roughly the status quo, underlining the trends that we saw developing at the general election as the political parties grappled with a new landscape after the referendum.
Polling expert Professor Sir John Curtice said the Tories were performing relatively well in areas with a substantial Leave vote in the 2016 referendum, where they are picking up votes from UKIP, which did not stand candidates in many areas, while Labour were performing better in places where the Remain vote was stronger and with a higher proportion of younger voters.
What else is happening?
In addition to the council polls, there have been a series of mayoral elections in London boroughs. Labour mayors were elected in Hackney, Lewisham, Newham and Tower Hamlets - while in Watford, the Lib Dem candidate was elected. Labour MP Dan Jarvis has been elected mayor for the Sheffield City Region.
There are no local elections in Scotland, Wales and Northern Ireland. However, a parliamentary by-election for the Westminster seat of West Tyrone took place, with Sinn Fein retaining it.