The new Taoiseach has briefed members of his Cabinet on the potential impact and preparations for Storm Éowyn.
Fianna Fáil leader Micheál Martin will also begin preparations for establishing the new unit on disability that will sit in the Department of the Taoiseach.
Over the coming days, he will meet the ministers and secretary-generals to outline the priorities for the coming weeks and months.
On Monday he will attend a ceremony to commemorate 80 years since the liberation of Auschwitz.
It comes after Mr Martin named his new Cabinet.
Fine Gael leader Simon Harris will be Tánaiste and Minister for Foreign Affairs and Trade as well as Minister for Defence.
Fianna Fáil TD Jack Chambers is moving from finance to take up the role of Minister for Public Expenditure, Infrastructure, Public Services, Reform and Digitalisation.
Read more: Cabinet list in full with number of promotions, changes
Paschal Donohoe becomes Minister for Finance once again.
Fianna Fáil TD Jim O'Callaghan, who is also a barrister, is the new Minister for Justice.
The previous minister for justice, Fine Gael's Helen McEntee, is the new Minister for Education and Youth.
Fine Gael TD Jennifer Carroll MacNeill will serve as the new Minister for Health.
Fianna Fáil's Dara Calleary will be Minister for Social Protection and Minister for Rural and Community Development and the Gaeltacht.
Fianna Fáil TDs Norma Foley and Darragh O'Brien, who were in the last cabinet, are moving departments.
Ms Foley, who was the minister for education, will take up the role as Minister for Children and Disability and former minister for housing Darragh O'Brien is moving to the Department of Transport, Environment and Energy.
Fianna Fáil TD for Waterford Mary Butler will take up the role of Government Chief Whip and will also have responsibility for Mental Health.
Fianna Fáil's James Lawless will be appointed as Minister for Higher Education Research, Innovation and Science and Fine Gael's Patrick O'Donovan will be Minister for Arts, Culture, Communications, Media and Sport.
Fine Gael’s Martin Heydon replaces Charlie McConalogue as Minister for Agriculture, Food, Fisheries and the Marine.
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Senior counsel Rossa Fanning has also been reappointed as Attorney General.
The announcements mean that of the eight ministerial positions for Fianna Fáil including Taoiseach, three TDs - Mr Lawless, Mr Browne and Mr Calleary - have been promoted to full ministries.
However, the only minister to keep their old portfolio is Peter Burke, who remains as Minister for Enterprise and Employment.
Three super junior ministers, who will be allowed to sit at Cabinet but not allowed to speak at it, have also been appointed.
They are Fine Gael TD Hildegarde Naughton, and Regional Independent TDs Sean Canney and Noel Grealish.
Among the other TDs who were rumoured to be in contention for a promotion to cabinet included Fianna Fáil's Niamh Smyth and Timmy Dooley, both of whom have lost out on a senior ministry role.
However, a number of junior ministry positions are available, with the new Government increasing the number of junior ministries from 20 to 23.
After naming his Cabinet, Mr Martin said the Government will work "every day to protect Ireland".
It will take a "whole of government approach taking key issues out of departmental silos and delivering action on multiple fronts at once".
"A core priority for this government" will be protecting "Ireland's position within the European Union, and in the wider international forum", he added.
Read more:
Read Micheál Martin's full speech on nomination as Taoiseach
Who has been appointed to the new Cabinet?
Earlier, the Dáil voted by 95 votes to 76 to elect Mr Martin to the office of Taoiseach.
Mr Martin was proposed for the role by Fianna Fáil TD Albert Dolan and seconded by Catherine Ardagh.
Mr Martin then travelled to Áras an Uachtaráin to receive his Seal of Office from President Michael D Higgins.
Sinn Féin leader Mary Lou McDonald was also nominated for Taoiseach in the Dáil.
She was proposed by Deputy Mairéad Farrell and seconded by Pearse Doherty.
Regional Independent Group agreement
The process of electing the Taoiseach got under way after an agreement was reached between the Government and opposition parties to resolve the dispute which was derailed yesterday's proceedings.
After talks which ran into the early hours of this morning, Ceann Comhairle Verona Murphy ruled that the Regional Independent Group would play no part in the Dáil today.
This issue of their speaking rights from the opposition benches saw the Dáil being adjourned four times yesterday.
This morning Ms Murphy said that under the temporary arrangement, the Regional Technical Group would be excluded from today's proceedings.
She said this issue will continue to be resolved following today's sitting until the Dáil returns on 5 February.
Government Chief Whip Hildegarde Naughton said there was "ambiguity" around the rules and the matter would now be examined by a committee.
The nub of the problem stemmed from the decision by four Independent TDs who intend to support the Government in all votes - Michael Lowry, Barry Heneghan, Gillian Toole and Danny Healy-Rae - to join a Regional Independent Technical Group and secure speaking rights from the opposition benches.
Ms Murphy suspended the House on several occasions and ultimately adjourned proceedings.
Last night, opposition parties wrote to her about a possible compromise, and this led to several in-person exchanges between Ms Murphy and Sinn Féin leader Mary Lou McDonald.
A letter was sent to Mr Harris and Mr Martin, which opposition leaders believed could be the basis of a compromise.
It sought to expand the Dáil's Standing Order 170 to exclude, under three different criteria, any TD supporting a government from obtaining speaker rights from opposition benches.
The criteria, included in a letter to Coalition leaders, exclude a TD from opposition speaking rights if they: Negotiated a Programme for Government, voted supporting a Programme for Government, or indicated they would support a Programme for Government.
Opposition leaders felt this proposal could end the stalemate.
Additional reporting Mícheál Lehane
