Here, we have a look at the topics likely to dominate political discourse in the week to come.
Speaking rights row a sign of a more united opposition?
The theatrics in the Dáil on Wednesday made international news, and denied Micheál Martin's family the chance to see him named Taoiseach for a second time.
The opposition had a good reason to be angry as the debate over non-ministerial Regional Independent TDs being given speaking time rumbled on.
They were furious at Ceann Comhairle Verona Murphy ruling that the Independents, supporting the new Government, would be given opposition speaking time.
Some of those independents are to be given jobs as Ministers of State while the remainder want to join a “technical group”, which is a mechanism designed to allow opposition TDs to sit in groupings of at least five members to gain an allocation of speaking time.
Ms Murphy was forced to suspend the Dáil, before it returned on Thursday to see Mr Martin become Taoiseach and the new Cabinet named.
Wednesday's event raises two issues. The first is whether Ms Murphy will be able to regain her authority over a Dáil that is sure to see more heated rows.
The other is whether this is a sign of a more united opposition.
Sinn Féin emerged as the second- largest party in the country after the general election, but the party is essentially locked out of government as Fine Gael and Fianna Fáil rule out working with them consistently.
The lack of any viable alternative government has been cited as a big issue, as the likes of Sinn Féin, Labour, the Social Democrats and People Before Profit have too many fundamental differences.
In a recent interview with BreakingNews.ie, DCU Politics Professor Gary Murphy said a more united opposition may be the only way for an alternative government to emerge.
Prof Murphy said: "Labour will want to rebuild, the Social Democrats will want to push on, it's a bit different maybe for Sinn Fein, but I do see the government lasting.
"For Sinn Féin, they must still be thinking 'where did it all go wrong?', they were consistently at 33 per cent for over a year.
"Fianna Fáil had already ruled it out [a Sinn Féin coalition], then you were looking at a left alliance from the radical People Before Profit at one end of the spectrum to the more moderate Labour at the other.
"Going forward if this government remains in office for five years, the way to get it out will be an alignment whereby Sinn Féin, Labour, the Social Democrats will have to say 'look, this is the alternative government'."
He added: "That will mean working together during this Dáil and then presenting themselves as 'we want you to vote for Sinn Féin, but also to transfer to Labour and the Social Democrats' and vice versa.
"If it remains as disparate as it was in the election just gone, I don't think Fianna Fáil and Fine Gael will be going anywhere at around 20 per cent... so getting them out will only work if there is much more cooperation and probably some sort of voting pact."
Lack of gender balance in Cabinet
The Government has been criticised for the lack of gender balance in the new Cabinet.
Fine Gael’s Helen McEntee was appointed Minister for Education and Youth, Jennifer Carroll MacNeill of Fine Gael is the new Minister for Health, while Fianna Fáil’s Norma Foley has taken up the role of Minister for Children and Disability.
Fianna Fáil's Mary Butler has been appointed as the Chief Whip, which means she will also attend Cabinet meetings.
Sinn Féin leader Mary Lou McDonald and Labour leader Ivana Bacik have been among those criticising the lack of women in the new Cabinet.
Junior ministeries to be named
There will be a record 23 junior ministers in the new Government, a move that has also been criticised by oppositions TDs.
This will include super-junior ministries for Seán Canney and Noel Grealish from the Regional Independent Group, and Minister of State positions for Michael Healy-Rae, Marian Harkin and Kevin “Boxer” Moran.
Mr Harris and Mr Martin will have some tough calls to make as they look to promote TDs, some of whom were mentioned for Cabinet positions.
Those who could be in line for junior ministeries include Fianna Fáil's Niamh Smyth and Jennifer Murnane O'Connor, and Fine Gael's Alan Dillon, Emer Higgins and Neale Richmond (who were previously junior ministers).
A new position created will be a Minister of State for migration, according to The Sunday Times.
Abroad
US president Donald Trump is continuing to issue executive orders in his first week back in office.
Mr Trump has ended his predecessor’s hold on sending 2000-pound bombs to Israel.
The move lifted a pressure point that had been meant to reduce civilian casualties during the US ally’s war with Hamas in Gaza that is now halted by a tenuous ceasefire.
In the UK, Labour chancellor Rachel Reeves is working to help businesses hit by her tax increases following the recent budget.
Meanwhile, prime minister Keir Starmer is to speak with Mr Trump this week, and is likely to encourage the president not to hit the UK with his promised tariffs.
