One of the probes will examine whether there was a cover-up at the Home Office regarding claims of abuse by Westminster politicians in the 1980s, contained in a dossier handed over at the time by the former Tory MP Geoffrey Dickens.
The other, involving a Hillsborough-style panel, will look into whether public bodies failed in their duty of care towards children in such cases as the Jimmy Savile scandal, the North Wales child abuse allegations and the grooming of young girls in Rochdale, Derby and Oxford.
This will have access to all government papers, including intelligence dossiers, and could call witnesses subject to the constraints of any criminal probes. If thought necessary, it could be upgraded to a full public inquiry.
After days of saying a wide-ranging inquiry was not needed, the political and media pressure appear to have forced David Cameron's hand. The Prime Minister, who No 10 said was taking a close personal and detailed interest in the claims, insisted he was determined to arrive at he truth, promising there would be no stone unturned.
Over the weekend he met senior colleagues including Home Secretary Theresa May. It is thought their change of heart was partly prompted by the stark comments of Lord Tebbit, who was at the top of government through most of the 1980s, that there may well have been a cover-up over allegations of a paedophile ring involving senior politicians.
The two government investigations are in addition to probes by the police into claims of child abuse at a London guest house, an inquiry into Savile's connections with the NHS and two probes by the BBC into the activities of the jailed TV entertainer Stuart Hall.
Ms May announced that Peter Wanless, head of the NSPCC, would lead the review of the Home Office's own investigation of papers alleging abuse. It emerged at the weekend that more than 100 of these have disappeared. The Wanless Inquiry will take eight to 10 weeks and also look at how the police and prosecutors handled information given to them.
A senior public figure will chair the panel looking at the wider issue of how public bodies dealt with child abuse allegations.
The Home Secretary said this inquiry would take much longer and probably not report back until after the 2015 General Election.
In a Commons statement, Ms May told MPs: "I want to address two important public concerns. First, that in the 1980s the Home Office failed to act on allegations of child sex abuse and, second, that public bodies and other important institutions have failed to take seriously their duty of care towards children."