The 33rd Dail is sitting for the first time on Thursday, some 11 days after the new political landscape became clear.
The 160 TDs gathered at Leinster House where all their names were read into the record by the Clerk of the Dail Eireann.
Procedures got under way at noon with Clerk Peter Finnegan reading out the Proclamation.
Fianna Fail TD Sean O Fearghail was re-elected as Ceann Comhairle in the Dail after receiving 130 votes.
He beat Independent TD Denis Naughten who received 28 votes.
He was elected following a secret ballot.
Mr Naughten was the first to make his pitch for the role.
He cited his experience as a parliamentarian at all levels, from TD to cabinet minister.
Mr Naughten said there is “public frustration” over accountability in public services.
“Getting straight answers to straight questions has become very difficult in Dail Eireann,” he said.
Before he was elected Mr O Fearghail told the Dail that he was seeking a renewal of their support for the role.
“I work in the basic principle of respect for the mandate that each member of the Dail has received from their electorate,” he said.
“That mandate is in every respect equal and deserving of similar priority.”
Ballot papers were given to TDs with the secret ballot taking place in the voting lobbies in specially constructed booths.
One of his first jobs as the Ceann Comhairle was to seek nominations for the position of taoiseach.
Sinn Fein leader Mary Lou McDonald, Fianna Fail leader Micheal Martin, Fine Gael leader Leo Varadkar and Eamon Ryan were all nominated for the role.
However, a taoiseach is not expected to be elected as there is no party with a majority.
While Ms McDonald is expected to shore up the most votes in the Dail, it is expected she will fall short of the 80 votes needed to win.
Ms McDonald received support from Solidarity-People Before Profit, Independents 4 Change TD Joan Collins, as well as Independent TDs Catherine Connolly and Thomas Pringle.
TD Richard Boyd Barrett said that Solidarity-People Before Profit’s vote for Ms McDonald was not giving her party a “blank cheque”.
“We think the only way to give effect to that clear demand of people is to establish a left minority government and the only realistic way of doing that would involve supporting Mary Lou McDonald as taoiseach and ruling out any deals with Fianna Fail or Fine Gael to form a government,” he added.
Ms McDonald earlier said that Sinn Fein will continue to speak to the parties who have a mandate for change.
“I also said that we will talk to everybody because that’s what you do in a democratic system, that’s what grown-ups do,” she said.
“I think it is a matter of great regret that Fianna Fail, in particular, continues to maintain a stance of excluding or ignoring or seeking to marginalise Sinn Fein, and not just us as elected representatives, but much more importantly, the hundreds of thousands of people that we represent.
“I don’t think that’s an acceptable position.
“I don’t think it’s a sustainable position, I don’t think it’s a democratic position for them to hold, and we have to see in the coming days and weeks whether or not they hold to that stance.
“I do know this much, I know that nobody voted for a return of Fianna Fail and Fine Gael.
“I also know from speaking to people out and about, people of all politics and none, that any suggestion of a return of the old Fianna Fail and Fine Gael government will cause huge disappointment right across the country and, I imagine, huge anger.”