The Irish Prime Minister (Taoiseach), Enda Kenny, has formally apologised on behalf of the state for its role in the Magdalene laundries.
Some 10,000 women and girls were made to do unpaid manual labour in laundries run by Roman Catholic nuns in Ireland between 1922 and 1996.
More than a quarter of those who spent time in the laundries had been sent there by the Irish state.
Mr Kenny apologised to all the women affected.
He said their experiences had cast a "long shadow" over Irish life and that it had been "humbling and inspiring" to meet them.
"For 90 years Ireland subjected these women, and their experience, to a profound indifference," he said.
"By any standards it was a cruel and pitiless Ireland, distinctly lacking in mercy.
"We swapped our public scruples for a solid public apparatus."
The Irish government is to ask the Law Reform Commission's president, Judge John Quirke, to carry out a three-month review to recommend the criteria for providing support and payments to the survivors.
The terms of reference were published on Tuesday evening.
An emotional Mr Kenny concluded his speech to a standing ovation in the Irish parliament.
The ovation was given to the survivors who were present in the public gallery.
Mr Kenny concluded his speech by referring to a song Whispering Hope that had been sung by one of the women present in his recent meetings with the survivors.
He said one of the lines from the song had remained in his mind.
"When the dark midnight is over, watch for the breaking of day," he said.
One of the survivors, Mary Smyth, said Mr Kenny's emotion had shown he believed them.
Steven O'Riordan of Magdalene Survivors Together said he was pleased that compensation had been proposed.
Justice for Magdalenes said it welcomed Mr Kenny's apology on behalf of the Irish state.
It said it looked forward "to the intent of the apology being made evident by the introduction of a system of redress that is prompt, open, fair, and transparent".
Earlier this month, Mr Kenny expressed regret for the stigma and conditions suffered by women who were inmates in the laundries, but he stopped short of a formal government apology.
In the two weeks since, Mr Kenny has held face-to-face meetings with some of those who worked in the laundries.
Mr Kenny said on Tuesday that the state apology and any Irish government fund would extend to residents of laundries not included in the McAleese report.
The Irish deputy Prime Minister (Tanaiste), Eamon Gilmore, said he wanted to tell the survivors that "we have heard you, we believe you, and we are profoundly sorry for what was done to you".
An inquiry chaired by Senator Martin McAleese found more than 2,000 women and girls were sent to the laundries by the state authorities, and many Irish institutions, such as the army and some government departments, had contracts with the laundries.
Women were forced into Magdalene laundries for a crime as minor as not paying for a train ticket, the McAleese report found.
The report also confirmed that a police officer could arrest a girl or a woman without warrant if she was being recalled to the laundry or if she had run away.
Fianna Fail has called for the establishment of a dedicated unit within the Department of Justice to co-ordinate the Irish Republic's response to the McAleese report, including all forms of redress for the survivors.
Its leader Micheál Martin said on Tuesday that he acknowledged the failure of all who had participated in public life in the past and had not acted to "intervene sooner" to apologise to the survivors.
He said that as a member of the previous government, he was "sorry" that had not happened "over the past decade".
The system was the subject of a 2002 film, The Magdalene Sisters, which starred Geradline McEwan and Anne-Marie Duff, whose director said at the time he believed the former inmates should have received an apology.
Earlier on Tuesday, Amnesty International accused the Irish government of ignoring women who were exploited in laundries that operated across the border in Northern Ireland.
Patrick Corrigan, the human rights charity's Northern Ireland director, said: "Magdalene laundries operated in Northern Ireland into the 1980s.
"I have spoken with women survivors of these institutions who now fear being left behind, with no inquiry in place - north or south - into their suffering."
The Good Shepherd Sisters ran laundries in Belfast, Newry and Londonderry.
Another Magdalene asylum, including steam laundry, was operated by the Church of Ireland on Belfast's Donegall Pass.