Taoiseach Micheál Martin has said it is "incumbent on all political parties" to form an Executive as counting in the Northern Ireland election draws to a finish.
Mr Martin congratulated all of the successful candidates on their election and said it is now incumbent on them to deliver on their mandate through the nomination of a first and deputy first minister and the formation of a new Executive.
"Power-sharing and principles of partnership, equality and mutual respect are at the heart of the Good Friday Agreement, through which peace has been secured and progress achieved for almost 25 years.
"A new power-sharing Executive is vital for progress and prosperity for all in Northern Ireland.
"As a co-guarantor of the Good Friday Agreement, the Government will continue to work in partnership with the British government and engage with the leaders of the political parties in Northern Ireland to seek and support the effective operation of all of the political institutions."
Tánaiste Leo Varadkar said that the big winner in the election was the Alliance Party.
He said the percentage of people who voted for nationalist and unionist parties has gone down.
Mr Varadkar said there is a growing middle ground of people who do not want to be defined by religion or ethnic or national identity and "to me that is really encouraging and really presents an opportunity for a new Northern Ireland into the future".
The Minister for Foreign Affairs described the election in Northern Ireland as historic.
Simon Coveney said Sinn Féin has undoubtedly been the big winners and he said it was significant that "a nationalist party now become the largest party".
He also noted the performance of the Alliance Party, which he said was also a big message coming from the electorate in the North.
"We're seeing a middle ground in Northern Ireland who don't want to be categorised as unionist or nationalist but want a middle ground politics to strengthen."
Speaking in Dublin, Mr Coveney said the days ahead will be challenging for all the political parties in Northern Ireland and for both the Irish and British governments.
He admitted the process of "putting together shared government in an Executive won't be easy", following what he described as "a very polarising election".
In the United States, the chairman of the Congressional Friends of Ireland has called on all the parties in Northern Ireland to work together to deliver a power-sharing government.
Representative Richard Neal said the symbolic importance of the results cannot be overstated, but said any form of triumphalism should be resisted, in order to protect the system of power-sharing.
The chairman of the House Ways and Means committee said "the preservation of peace and stability remain paramount, regardless of the outcome of these elections. Now is not the time for hard lines that will diminish this great progress".