The Scottish Government has announced plans for an independent team to look at individual cases in the scandal in the hope of finally providing relatives with some answers.
It follows a long-awaited report on the issue. It made 64 recommendations, including changes in the law for crematoria.
Scottish Public Health Minister Michael Matheson yesterday accepted the report by the independent Infant Cremation Commission "in full, and without reservation", before confirming the introduction of a national investigation team, headed by Dame Elish Angiolini.
Patrick McGuire, from Thompsons Solicitors, who represents more than 200 families, said: "This announcement is very welcome. It is what the families I represent have been asking for.
"The families are particularly pleased it will look closely at each individual case and, as far as possible, give every family the answers they are looking for.
"We are also reassured those people across Scotland who are identified as being responsible for this scandal will now be properly held to account."
The announcement came as the commission, headed by Judge Lord Bonomy, recommended a statutory definition of ashes be introduced alongside stricter regulations on the cremation of babies of less than 24 weeks gestation.
It also called for the introduction of a public record that would detail whether ashes were scattered, buried or collected by a family member.
The review was set up more than a year ago after it was revealed Mortonhall Crematorium, Edinburgh, had secretly buried the ashes of babies for decades without the knowledge of their families. Other local authorities have also subsequently been implicated.
Mr Matheson told MSPs all parents who have been left with unanswered questions about the cremation of their baby will now be able to have their case examined.
He said: "Dame Elish and her team will be able to look at every document and every record, they will interview every concerned family and will expect to speak to any officials or staff members who may hold information. They will be able to look at the NHS, at funeral directors, as well as crematoria. Parents can be reassured every step will be taken to find out what happened to their babies."
He said the publication of the commission's report, as well as an earlier review by Dame Elish into the Mortonhall ashes scandal, were "significant stepping stones".
However, he stressed: "There is much still to be done. There are new laws to make. There are procedures and processes to update. And there are individual cases and crematoria we will now investigate.
"Sadly, some parents will never know what happened to their children, but I hope those parents will recognise we will do all we can for them to get the answers that are available."
The commission's report recommends "the baby and the interests of the family should be the central focus" when devising policy and making arrangements for cremations.
It also suggested "ashes" should be defined in law as "all that is left in the cremator at the end of the cremation process and following the removal of any metal".
The Scottish Government is also recommended to set up a national committee with responsibility for baby and infant cremations and ministers are advised to appoint an independent inspector to monitor standards at crematoria.