In a widely expected announcement, Mr Robinson, 66, said he will not contest next May’s Assembly election and is likely to leave office in the coming weeks.
The move comes days after he signed a political deal with Sinn Fein and the Irish and British governments.
The head of the Democratic Unionists at Westminster will be one of the favourites to take over as leader.
Nigel Dodds is North Belfast MP and has served in the past as a member of the Northern Ireland Assembly and minister in the Stormont Executive.
The married father-of-two has been an upholder of the fundamentalist unionist party’s stance on social issues, supporting evangelical preacher Pastor James McConnell after he was accused of anti-Islamic slurs.
He has also argued strongly against the introduction of same- sex marriage.
Arlene Foster, a Fermanagh South Tyrone Stormont Assembly member, has stood in for Peter Robinson as first minister after he took a six-week hiatus in 2010 following a storm over his wife Iris’ affair and finances.
During recent political talks on paramilitarism as the powersharing government threatened to collapse, she also deputised as head of the administration.
The married mother is a lawyer from Co Fermanagh who joined the Democratic Unionists in January 2004, just weeks after resigning from the rival Ulster Unionist Party.
As a child she had first-hand experience of the violence that engulfed Northern Ireland during the Troubles.
The IRA tried to kill her father, a part-time policeman in the RUC while she was a schoolgirl.
As a teenager in 1988, she survived a bomb which exploded under her school bus which was being driven by a part-time Ulster Defence Regiment soldier.
Sammy Wilson, East Antrim MP, could be another leadership contender.
He has been a colourful presence in Parliament and the Assembly, known for his robust exchanges as Stormont finance minister and his unorthodox views on climate change as environment minister.
Another senior Democratic Unionist, Simon Hamilton, who has enjoyed a meteoric rise under Peter Robinson’s leadership, is seen as outside bet.
He is health minister in the Stormont Assembly having previously served as finance minister and is a moderniser keen to cut government waste.