Image copyright PA The first results are being declared in local elections in England, as counts continue across the UK.
Early results from a few councils in the North of England suggest Labour's vote is up on 2015 but down on 2012 when these seats were last contested.
The Tories are down on 2015 but doing better than in 2012, the Lib Dems are recovering slightly and UKIP is holding on to the vote share it won last year.
Results are due in overnight for the Scottish Parliament and Welsh Assembly.
The result of the high-profile London mayoral contest is not expected before Friday evening.
In Scotland, SNP leader Nicola Sturgeon is expected to lead her party to a third successive term in office, and there has been speculation that the Conservatives could push Labour into third place.
There are also mayoral contests in London, Bristol, Liverpool and Salford.
UK Parliamentary by-elections are also being held in Ogmore and Sheffield Brightside while police and crime commissioners are also being elected in England and Wales.
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Shadow Chancellor John McDonnell told Sky News that if Labour could reduce the 6.3% margin by which the Conservatives led in the 2015 general election they would "demonstrate steady progress".
He said the aim was to "steadily build our support" over four years and said the party had hit an "all-time high" when the seats were last contested in English councils in 2012. But he admitted conditions remained "tough" for Labour in Scotland.
Meanwhile, Shadow Home Secretary Andy Burnham is considering standing to become mayor of Greater Manchester in 2017, it has emerged.
Analysis
Image copyright PABy BBC Deputy Political Editor John Pienaar
The English local council seats up for grabs were won at the height of Labour's popularity under Ed Miliband - though that's not saying a great deal. Now, losses look inevitable, and a pantheon of psephologists predict 150, 190, more?
Mr Corbyn's Labour enemies have set the bar for respectability impossibly high.
Four hundred gains are the minimum to show progress, according to the likes of defeated leadership contender Liz Kendall, and given the fact Labour's trailing the Conservatives in the national polls, she might as well have demanded 400,000.
If Labour comes third in Scotland - though that's not thought very likely - or loses in London - something almost no-one expects to happen, the hair-rending and teeth-gnashing will be so frenzied as to leave the entire Labour Party, Corbynistas and anti-Corbynistas alike, bald and toothless.
As it is, we can expect a cacophonous set of rival choruses. Those loyal to the leader blaming the disloyalty of dissident MPs for vote-sapping disunity. And those who want the leader gone blaming him; demanding a change of direction and some a change of leader.
Read John's full analysis
BBC Newsnight's political editor Nick Watt said there was unlikely to be a leadership challenge against Mr Corbyn, unless Sadiq Khan loses to Zac Goldsmith in the London mayoral race.
In total, 2,747 seats in English councils - spanning metropolitan boroughs, unitary authorities and district councils - are up for grabs. The majority of the seats were last contested in 2012.
Earlier, problems were reported in Barnet, North London, where the council apologised after some people were unable to vote on Thursday morning.
Thursday's polls are the single largest test of political opinion before the next general election, which is scheduled for 2020, with 43 million people entitled to take part.
Image copyright Getty ImagesTap here to find out which election is taking place in your area.
Image copyright PANew MPs for the constituencies of Ogmore, in Bridgend County, and Sheffield Brightside and Hillsborough - both safe Labour seats - will also be chosen.
The Ogmore by-election was triggered by former Labour MP Huw Irranca-Davies's decision to stand for the Welsh Assembly. The Sheffield contest was caused by the death of sitting Labour MP Harry Harpham.
Voters in 40 police force areas in England and Wales, excluding London and Greater Manchester, will also elect a police and crime commissioner. The inaugural elections for the positions were held in 2012.
What to expect when (all timings are estimates)
Friday 6 May
00:30 First results from councils in England - around 80 results expected overnight
01:30 First results from Scottish Parliament - results in all 129 seats expected overnight
02:15 First results from Welsh Assembly - results in all 60 seats expected overnight
Liverpool mayoral result also expected overnight.
11:30 First results from around 50 councils in England to declare on Friday
15:00 First results of London Assembly elections
17:00 First preference vote share information for elections to Northern Ireland Assembly
1800 Result of first preference votes for Mayor of London (final result expected early evening)
Results of Police and Crime Commissioner elections in England and Salford mayoral contest are also expected on Friday.
Saturday 7 May
14:00 Results from five remaining councils in England
16:00 Result of Bristol Mayoral contest
Result of elections to Northern Ireland Assembly expected
Sunday 8 May
16:00 Result of elections to Bristol Council (final council in England to declare)
Results of Police and Crime Commissioner elections in Wales