DNA tests have proved that a seven-year-old girl taken from a Roma family in Dublin on Monday is their daughter.
The family said they were "delighted" that their daughter had come home.
They also said they would be taking legal advice, and that serious questions have arisen over the procedures used in the case.
The blonde-haired, blue-eyed girl had been removed by police from her Tallaght home and taken into temporary care.
The family have supported calls from human rights group Pavee Point for an independent inquiry into the investigation.
In a statement issued through their solicitor the family said: "Her removal has been a cause of huge upset to her parents, her brothers and sisters, and the young girl herself.
"They now intend to concentrate on looking after their family and, in particular, in trying to reassure their daughter that she will be left in their care.
"Our clients also wish to say they do not believe that what has happened to their family over the few days should ever have happened.
"They do not accept that this was any proper or sufficient basis to take their daughter away from them.
"They believe that there are very serious questions arising about the procedures used in this case but are going to wait for things to settle down and consider their position and that of their daughter in light of recent events and will be taking legal advice in respect of this."
A 21-year-old sister of the child, who can not be identified for legal reasons, said their mother had not eaten for three days because she was so distraught.
"Everyone was very sad," she said. The sister added she hoped no other family would have to go through a similar ordeal.
Meanwhile, a two-year-old boy from a Roma family who was briefly taken into care in County Westmeath has been reunited with his parents.
The boy was taken from his family on Tuesday in Athlone and returned a day later.
Alan Shatter, the Irish minister for justice, said he will be asking the Garda (Police) Commissioner for a report on the two cases.
A Garda statement said: "Protecting vulnerable children is of paramount importance to An Garda Síochána and we continue to work in partnership with the HSE (Health Service Executive) and other agencies to ensure children's safety.
"An Garda Síochána want to assure the community that we take extremely serious all reports received from members of the public concerning child welfare issues."
The Irish police action took place against the background of international interest in the case of a blonde-haired child being taken from a Roma family in Greece last week.
Greek police are investigating whether the girl had been abducted.