Sinn Féin president Mary Lou McDonald has described the Irish general election as “something of a revolution in the ballot box”.
Counting is continuing and the first results are coming in, after an exit poll put the three main political parties tied in first preference votes.
Indications suggest there is little difference in percentage terms between Fine Gael, Sinn Féin and Fianna Fáil.
Early tallies at count centres suggest a Sinn Féin surge.
Polling in the election closed at 22:00 on Saturday.
Ballot boxes from across the 39 constituencies were opened at 09:00 local time on Sunday.
Arriving at the RDS count centre in Dublin on Sunday afternoon, Ms McDonald said she was exploring options to see if it would be possible to form a government without either Fine Gael or Fianna Fáil.
Before the election both Fine Gael and Fianna Fáil said they would not enter coalition with Sinn Féin.
On Sunday evening the Taoiseach (Irish PM) and Fine Gael leader Leo Varadkar said it would be “challenging” to form a government.
He said his party has consistently ruled out forming a coalition with Sinn Féin “in large part because of their policies in relation to crime, tax and the way the economy and society should be run and also our deep concerns about their democratic structures.
“We don’t believe a coalition between Sinn Féin and Fine Gael is a viable option.
He said a “forced marriage would not result in a good government.”
But Fianna Fáil leader Micheál Martin did not rule out working with Sinn Féin, but said “significant incompatibilities” still existed.
He said reports suggested that his party “would be the largest”.
He added that he was a “democrat” and respected the vote of the people.
Ms McDonald said this election was about “change”.
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Media caption‘Excluding Sinn Féin would be wrong’
“The frustration people have felt for a long time with the two-party system, whereby Fine Gael and Fianna Fáil handed the baton of power between each other – that’s now over.”
“We now have a very substantial mandate,” she said.
In 2016 her party won 23 seats but projections for the 2020 election have them set for between 36 and 40.
The projection carried out by University College Dublin has the party on a similar number of seats to Fianna Fáil and Fine Gael.
It followed an exit poll – commissioned jointly by Irish national broadcasters RTÉ and TG4, as well as The Irish Times and University College Dublin – which indicated that Fine Gael secured 22.4% of first preference votes, closely followed by Sinn Féin (22.3%) and Fianna Fáil (22.2%).
Profile of Ireland’s political parties
Ireland’s elections are carried out under the proportional representation (PR) voting system, using the single transferable vote (STV). Voters wrote “1” opposite their first choice candidate, “2” opposite their second choice, “3” opposite their third choice and so on.
This means that the picture presented when the first preference votes are counted does not completely reflect the final outcome.
Sinn Féin also ran 42 candidates across the 39 multi-seat constituencies, about half that of both Fine Gael and Fianna Fáil, which will have a knock-on effect on the number of seats it can secure.
The exit poll also suggests the Green Party secured 7.9% of first preference votes, followed by Labour (4.6%), Social Democrats (3.4%), Solidarity People Before Profit (2.8%).
Indications are that Independents took 11.2% of first preference votes.
The poll suggests a move toward Sinn Féin among younger voters, with the party receiving the largest number of first preference votes among 18-24 years olds (31.8%).
The majority of voters over the age of 65 appear to have given their first preference to Fine Gael or Fianna Fáil.
There is a margin of error of 1.3% in either direction in the exit poll.
Political pundits have predicted that some government ministers could lose out, if early projections are correct.
RTÉ has said the potential casualties could include Transport Minister Shane Ross in Dublin-Rathdown, an independent who first won a Dáil seat in 2011.
Minister for Children, Katherine Zappone, is also in difficulty in Dublin South-West, the broadcaster said.
Former Labour leader and tánaiste (deputy prime minister) Joan Burton, first being elected to the Dáil in 1992, is also likely to lose her seat, RTÉ said .
Storm Mary Lou?
Analysis by Shane Harrison, BBC News NI Dublin correspondent
As Ireland north and south is battered by Storm Ciara, it is Storm Mary Lou that is the talking point of the Republic of Ireland’s election.
Mary Lou McDonald and Sinn Féin have ensured that what was once a predominantly two party system is no more as she and her colleagues capitalised on public concern about health care and housing.
But Sinn Féin did not run enough candidates to capitalise on its unexpected surge in support.
With both parties ruling out coalition with Sinn Féin, it may take some time to form a government.
And Mary Lou McDonald will need little reminder of how fickle the electorate can be.
In the past, the Labour Party surfed the mood of change with the 1992 Spring Tide and the 2011 Gilmore Gale: neither lasted very long.
But this time it might be different.
The electorate have spoken but it might take some time to figure out what voters have said.
RTÉ said voting appears to have been “solid”.
However, there is no expectation of a spike in voting compared to 2016, despite it being the first ever Saturday general election vote in the state’s history.
Factors that may have affected turnout include the poor weather and international rugby.
A total of 160 representatives will be returned to the Dáil (Irish parliament) and newly elected TDs will gather on 20 February .
The ceann comhairle, or speaker, is automatically re-elected.
In most situations, the speaker does not vote, so a government will need 80 TDs to hold a majority.
Polls open for first-ever Saturday general election vote
It is unlikely that any party will reach that number, so another coalition government is probable, although the composition is still unclear.