A new leader is to be in place no later than June 2nd
The Taoiseach Enda Kenny has stepped down as Fine Gael leader.
In a statement posted on Twitter earlier, Mr Kenny said he will continue to carry out his leadership duties in an acting capacity until a successor is elected.
"I want to assure people that throughout this internal process, I will continue to carry out my duties and responsibilities as Taoiseach in full", he said.
"I intend to provide a brief but appropriate period for my successor to engage with groups and members supportive of Government, and with other Parties in the Dáil regarding provision of Government for the future."
He also said he wants the leadership process completed by close of business on June 2nd.
Mr Kenny said it had been a "huge honour and privilege" to lead the party for the past 15 years.
"I would like to stress the huge honour and privilege that it has been for me to lead our party for the past 15 years, in opposition and into Government on two successive occasions.
"I thank all our members, past and present for that privilege.
"I thank all my loyal constituents and supporters in Mayo for their unstinting loyalty since 1975, and for their support for my family previously in providing unbroken service to the County in Dáil Éireann since 1954.
"I thank my personal staff for their commitment over many years to their duties.
"I especially want to thank my wife Fionnuala, our children, my siblings and their families for their understanding of my work, and indeed for accepting the many intrusions of politics into family life in the interest of building our Country. I could not have engaged as I did without that base."
He held a meeting of the Parliamentary Party on Wednesday, which was brought forward by an hour at the request of Mr Kenny himself.
This led to increasing speculation of an imminent leadership contest.
One of the favourites to succeed Mr Kenny as leader, Social Protection Minister Leo Varadkar, has paid tribute.
"Enda Kenny has been an extraordinary Taoiseach and Fine Gael's most successful leader ever. After decades of service to our party and country, we owe him a debt of gratitude", Mr Varadkar said.
"He brought Fine Gael from the edge of extinction to its current position as the largest party in the State. He brought Ireland from the brink of bankruptcy back on to the road to prosperity.
"When elected leader of Fine Gael in 2002, he brought renewal and hope to the party, which was at its lowest ebb.
"He immediately set about rebuilding. In eight years as leader of the opposition, his tireless work saw Fine Gael grow and gain seats in one election after another.
"Enda gave hope to a battered and bruised Ireland in 2011. His dedication and determination gave Fine Gael a landslide in that year's general election.
"But more importantly, he brought this country back from the brink of economic collapse, and offered political stability when countries across Europe were staring into the abyss."
Mr Varadkar also said he will "never forget the day that Irish democracy was handed back from the Troika, restoring our standing in the world and returning our dignity."
"It's incredible to think that we have climbed from those depths and are once again the fastest-growing economy in Europe", he added.
Mr Varadkar continued: "On a personal level, I owe him so much. He appointed me to the party's Front Bench following my first election as a TD.
"I was honoured to accept a ministerial position from him after the 2011 election. I have learned a lot from Enda, in particular how to lead a team of people with many talents and diverse views.
"Against all the odds and in spite of predictions to the contrary, Enda fashioned a Government in 2016.
"Once again Enda didn't flinch, and he brought stability to the Dáil, to Government, and to Ireland, at a time when there was no obvious solution. It was a privilege to be directly involved in those negotiations alongside the Taoiseach.
"Subsequently Enda faced probably the toughest challenge of his career: that of Brexit. He has tackled this head-on with vigour and determination, and has secured the best possible situation for Ireland in advance of detailed Brexit discussions.
"Enda's legacy will be far-reaching. Above all, he is a kind and decent man, always ready with an encouraging word or a solid piece of advice. He is devoted to family, party and country. I hope sincerely that his expertise will continue to be available to us. But I know that the party, and the country, will always remember him and will always be grateful."
Minister Varadkar and Housing Minister Simon Coveney have emerged as front-runners for the next Fine Gael leader.
The two men have varying views on several different aspects that will be crucial to any leadership win.
Tánaiste Frances Fitzgerald was asked Wednesday morning if there should be a third candidate in the race - and she refused to rule herself out.
She told reporters: "There is no vacancy at present - we have to wait and see what the Taoiseach has to say later this evening. I'll make my own position clear then."
Independent TD and Children's Minister Katherine Zappone, however, has given Minister Fitzgerald her encouragement to enter the leadership contest.
She observed: "I think she is a progressive voice within the party, she's a feminist, and I'd be very happy to see her throw her hat into the ring."
Mr Kenny was the longest serving Fine Gael Taoiseach. This landmark passed in April when he exceeded 2,234 days as leader.
This beat the record of John A Costello, who served two separate terms as Taoiseach in the 1940s and 50s.