Here, we have a look at the issues likely to dominate political discourse in the week to come.
Government formation in 'landing zone'
Taoiseach Simon Harris has said government formation talks are in the "landing zone".
Mr Harris said it may be "tight" to have a coalition in place before the Dáil resumes on January 22nd. However, he believes it can be done.
"We're in the landing zone of being able to deliver a new government," he said.
“People can work hard, work later, work longer – we’re doing that, and work through the weekend.”
The Fine Gael leader said forming a government within eight weeks of the election is “not an extraordinary length of time” when the Christmas break is taken into consideration.
Certain key areas of the programme of government are understood to have been largely agreed between the two parties, although talks on disability, justice and climate continue.
Rotating taoiseach to return
After Fianna Fáil won 10 more seats than Fine Gael in the general election, there were questions over whether the rotating taoiseach arrangement would continue.
After a busy nine months in the office, Mr Harris will be targeting another spell as taoiseach, and it appears he has got his way in negotiations as he confirmed the agreement would return.
Asked if Fine Gael would enter into Government without a return to the rotating Taoiseach arrangement, Mr Harris said: “There will be a rotating Taoiseach in the next government and I think that’s been very clear probably since the day or two after the election.”
Mr Harris said the arrangement had worked well in the previous government.
Speaking on Tuesday, he added: “I believe many hundreds of thousands of people in this country have voted for Fine Gael and Fianna Fáil to provide stable government for the next five years, and for myself and Micheál Martin to lead that government.”
Harris has eyes on foreign affairs?
If Mr Harris hopes to succeed Micheál Martin as taoiseach, he will have to keep a high-profile, and he may well be targeting the role of Minister for Foreign Affairs.
He will undoubtedly take a senior ministerial role along with the position of tánaiste, and he may well be considering how Mr Martin has benefitted from the role.
After Fianna Fáil secured 48 seats in the general election, his position as leader of the party is very secure.
However, there was a wobble a couple of years ago, with consistent murmurings of discontent among backbench TDs. Some speculated that Mr Martin could suffer from being abroad so much, with rumours of secret meetings when he was away.
However, having such a prominent role on the world stage has served him well, and it may well be tempting to Mr Harris.
He has been linked with the departments of enterprise and justice (which he served in temporarily while Helen McEntee was on maternity leave), but his recent comments in an opinion piece in The Irish Times suggest he has his eyes on foreign affairs.
Michael Lowry's role
Speaking this week, Social Democrats deputy leader Cian O'Callaghan confirmed the party would not enter into government after talks with Fianna Fáil and Fine Gael.
This means it is all but certain the government will make a deal with the Regional Independent TDs Group.
The Regional Independent Group is made up of Michael Lowry, Sean Canney, Kevin 'Boxer' Moran, Noel Grealish, Marian Harkin, Barry Heneghan, Gillian Toole and Verona Murphy.
Michael Lowry is understood to be one of the leaders in the group, and was a key voice in the ultimately successful nomination of Ms Murphy as Ceann Comhairle.
Mr Lowry's past has been the subject of some criticism amid the negotiations.
The Moriarty Tribunal was set up in 1997 to examine payments made to former taoiseach Charles Haughey and Mr Lowry.
It found Mr Lowry's behaviour was "profoundly corrupt".
Asked if he had any hesitation about including Mr Lowry in a future government, Mr Harris said he fully accepted the findings of the tribunal.
However, he added: “I also know the practicality of how the Dáil works, and that when people get elected to Dáil Éireann, they can choose to support or not support governments – that’s the way our parliamentary democracy works, and the deputy has been elected.
“But my issue here isn’t about individuals, it’s about recognising that there are a number of Independent TDs who have come together in a grouping called the Regional Independents.
"They have indicated that they wish to try and support a government, and we’re happy to engage with them on that basis.”
Healy-Raes pushing for ministerial role?
Kerry Independents Danny Healy-Rae and Michael Healy-Rae have indicated they could support the government in votes, further strengthening their majority.
The pair have met Micheál Martin on a number of occasions and indicated they would be willing to back the government in return for assurances about certain supports for their constituencies.
Speaking recently, Michael Healy-Rae said: “The government has made it very clear they want us and they need us. We were asked if we were willing to support them for five years and the answer was yes provided that Kerry was looked after.”
However, the brothers are reportedly pushing for a ministerial role which could complicate negotiations.
A super-junior ministerial position would have to be on offer for the two Healy-Rae brothers to strike a deal with Fianna Fáil and Fine Gael to enter a coalition government, sources close to the two Kerry TDs told The Irish Times on Tuesday.
Abroad
Elon Musk continues to make waves in UK politics.
The UK Home Office minister suggested her life had been turned upside down and the risk to her safety had risen after Mr Musk used his X social media platform to accuse her of being a “rape genocide apologist”.
Mr Musk “knows absolutely nothing” about the subject of grooming gangs, Jess Phillips has said, after the billionaire attacked her on social media.
Political circles in the UK had speculated Mr Musk was preparing to give Nigel Farage's Reform UK party $100 million (€96 million) in funding.
However, Mr Musk's relations with Mr Farage have soured as he suggested he should be removed as Reform leader. This is because Mr Farage refused to back Mr Musk's demands for the release of far-right agitator Tommy Robinson.
Robinson, real name Stephen Yaxley-Lennon, is currently serving an 18-month prison sentence for contempt of court.
In the US, president-elect Donald Trump made some alarming comments at a press conference this week.
He said he would not rule out the use of military force to seize control of the Panama Canal and Greenland, as he declared US control of both to be vital to American national security.
Mr Trump also warned "all hell will break loose" if the Gaza hostages aren't released by his inauguration.
