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%.