Dame Elish has been appointed by Edinburgh City Council after calls from parents whose infants' remains were buried without their knowledge after being told there was none at Mortonhall Crematorium in Edinburgh.
The first woman to take the top law officer's post in Scotland, Dame Elish now holds a key role as Principal of St Hugh's College at Oxford University.
It is anticipated Dame Elish will review the files, conduct further investigation and interviews and draw on other expert advice.
The current review by independent auditors PriceWaterhouseCoopers into the crematorium records will feed into Dame Elish's investigation, as will the policy review activity being led by the council's services for communities directorate in consultation with bereavement charity Sands Lothian and others.
Dorothy Maitland of Sands said: "I am very happy with the appointment. I think they have taken it very seriously."
She said she had no concerns at this stage with any potential clashes with Dame Elish's role at Oxford. "Already she has made contact with people so at this stage I feel quite confident she is the right appointment."
Dame Elish will report to chief executive Sue Bruce.
The findings of the investigation will be reported to the council's Transport and Environment Committee, which is led by Lesley Hinds.
The start date of the probe will be contingent on clarification from Lothian and Borders Police of the status of complaints made to them in relation to Mortonhall Crematorium and to ensure any police investigation is not impeded.
Ms Hinds said "As a former Lord Advocate, Dame Elish is held in the highest esteem and reflects the authority and experience required to properly investigate this significant and sensitive matter. I look forward to receiving her findings."
A police spokesman said: "Lothian and Borders Police can now confirm a total of 19 complaints have been received in relation to Mortonhall Crematorium in Edinburgh.
"All of these complaints will be thoroughly assessed in order to determine whether any further police action is appropriate."
The Scottish Government has said a full public inquiry may be necessary after the Dame Elish investigation.
The investigation, which may be subject to amendment, will review: Initial findings; the audit of crematorium records which is currently under way; the council's arrangements for communicating with all parents who have registered an inquiry and who are to receive a full response to their inquiry supported by the relevant documentation; whether evidence from the records audit, the initial findings and any other interviews indicate failures in professional standards and/or management practices; current policy and practice recommended in the attached report and contribute as necessary; com-munication process between Mortonhall, NHS Lothian, funeral directors and bereaved parents; national guidance and policy and practice in other authorities in order to establish whether guidance within Edinburgh requires to be reviewed and refreshed; and any other relevant matters.
Kenny MacAskill, the SNP MSP for Edinburgh Eastern who is also the Justice Secretary, said: "The fact Dame Elish Angiolini has been appointed to lead on this demonstrates the high importance of this investigation and the sensitivity needed to conduct such an inquiry.
"As former Lord Advocate, I know Dame Elish well and hold her in the highest esteem. I am keen to see her findings."