Fianna Fáil and Fine Gael's negotiating teams have concluded their first talks on the formation of a minority Government.
It is understood the discussions went well, however the exact details of the meeting have yet to be revealed.
Speaking this evening the Taoiseach said elected representatives have to be mindful of their responsibilities to those who elected them.
The meeting has been described by both sides as constructive.
It was held in the Sycamore Room in government buildings and lasted just 90 minutes.
Afterwards, the two parties issued identical statements describing the encounter as "constructive and cordial".
The Fine Gael team was made up of Frances Fitzgerald, Leo Varadkar, Pascal Donohoe and Simon Coveney.
Fianna Fáil was represented by Michael Mcgrath, Charlie Mconologue Barry Cowen and Jim O 'Callaghan.
Both parties have agreed to meet again tomorrow.
The Dáil will vote for a new Taoiseach for a third time this Thursday.
It remains to be seen whether that will give the parties enough time to agree a deal that would see Enda Kenny become the first Fine Gael leader to be re-elected as Taoiseach.
It is expected the parties will agree a framework of policies that could be implemented in a Fine Gael-led minority Government.
The meeting is being welcomed by Independent Michael Healy Rae: "I hope that the talks are going successfully and that they will conclude in the very near future, hopefully by tomorrow or Wednesday and an agreement will be put in place.
"And, for God's sake, at the end of the day, that the people will get what they deserve and that's a Government and politicians working properly on their behalf."
The Independent Alliance is meeting in the morning to decide how it will vote on Thursday.
Alliance member Finian McGrath says the two biggest parties need to come up with a durable agreement: "I’m glad that we forced them into having these talks so they can sit down and hammer out some sort of an arrangement, because we need to know if that we back a particular side that we need to know we’ll get at least three years out of it."
Meanwhile the new Government - whatever its make-up - is being reminded there's a long list of items to be tackled.
Former Tánaiste and Justice Minister Michael McDowell says there are important issues on the horizon which can't be ignored: “There’s industrial relations problems coming down the tracks big time for us.
“The so-called fiscal space seems to have evaporated with increasing costs in the health sector and the like.
“All of these pre-election promises about cutting and slashing USC and abolishing it, that seems to have all disappeared off the table.
“And we have the possibility of Brexit,” he concluded.