His government has lasted just more than a year, with limited support from the main opposition party, Fianna Fáil.
Brexit and Northern Ireland's political stalemate are among the challenges waiting for the new taoiseach.
The leadership contest will begin on Thursday, with the new leader to be announced on 2 June after an electoral college of the parliamentary party.
County councillors and party members will also have their say.
While the field may widen before 2 June, the early favourites for the post are Social Protection Minister Leo Varadkar and Housing Minister Simon Coveney.
Born in Dublin, the 38-year-old is the son of a Mumbai-born doctor and a mother from County Waterford.
He was privately educated and studied medicine at Trinity College in Dublin, qualifying as a GP before entering politics.
He was elected to parliament in 2007 as a representative for Dublin West and held the tourism and health portfolios before taking on his current post last year.
In an RTÉ interview in 2015, he revealed that he is gay, explaining that he wanted to be clear on the matter ahead of the Irish referendum on same-sex marriage.
"I just kind of want to be honest with people," he said. "I don't want anyone to think that I have a hidden agenda."
Mr Varadkar has courted controversy by announcing that people convicted of welfare fraud will be "named and shamed" on a government website.
In a recent speech reported by the Irish Independent, he outlined his attitude to Brexit, saying there should be "no economic border at all between north and south".
The son of the former Fine Gael politician Hugh Coveney, the Cork-born 44-year-old has followed his father's footsteps to a seat in the Dáil (Irish parliament).
As a teenager, he was expelled from the exclusive Clongowes Wood boarding school for drinking and bunking off to attend a beach party.
In a candid interview with RTÉ, he said he had suffered from a stutter, and lived in the shadow of his more promising brother, who is now a successful businessman.
He went on to study management at the Royal Agricultural College in Gloucestershire, England, before his election as the youngest member of the Dáil in 1998.
He was appointed as agriculture minister in 2011 and responsibility for defence was added to his portfolio three years later.
He has made no secret of his desire to lead his party - when RTÉ asked the question about his intentions, he said: "I am very ambitious, so the straight answer is yes, some day.
"But I am personally very loyal to Enda and he knows that."
As Irish PM Enda Kenny quits as Fine Gael leader, eyes are focusing on who might succeed him.