The Met Police and NSPCC outlined offences at 13 hospitals, including Great Ormond Street in London and Wheatfields Hospice in Leeds.
Some 214 crimes were recorded across 28 police force areas, including 34 of rape or penetration, the report said.
The CPS apologised for missing the opportunity to prosecute Savile in 2009, while he was still alive.
Police said the victims’ accounts painted a “compelling picture of widespread sexual abuse by a predatory sex offender,” and Cdr Peter Spindler, who is leading the abuse probe, said Savile had “groomed the nation”.
David Cameron’s official spokesman said it was “absolutely right that every institution involved gets to the bottom of what has gone on”.
The former BBC presenter and Radio 1 DJ died aged 84 in October 2011, a year before the allegations emerged in an ITV documentary.
Revelations that Savile had sexually abused children prompted hundreds of victims to come forward, including those who said they were attacked on BBC premises and a number of other institutions.
In response to the report, the BBC said it was “appalled” some of the offences were committed on its premises and restated a “sincere apology to the victims of these crimes”.
The Giving Victims a Voice report set out the findings of Operation Yewtree, which launched three months ago to investigate the Savile abuse claims.
The 30-page document detailed a number of findings, including:
Savile offended at 13 hospitals, including Great Ormond Street, with one offence recorded at Wheatfields Hospice in 1977
A total of 14 offences were recorded in relation to schools
Savile’s youngest victim was an eight-year-old boy, and the oldest was 47
The earliest allegation is from 1955 in Manchester and the last is from 2009
Offences were carried out at the BBC between 1965 and 2006, including at the last Top of the Pops recording
Peak offending took place between 1966 and 1976
A total of 450 people have made sexual abuse allegations against Savile since October – of whom 73% of victims were under 18, with most aged 13 to 16
There is “no clear evidence” he operated in paedophile ring, although “whether he was part of an informal network” is still being investigated
The report also revealed 16 offences were committed at Leeds General Infirmary between 1965-1995 and 22 at Stoke Mandeville Hospital between 1965-88.
One offence was committed at each of the following hospitals – Broadmoor high-security psychiatric hospital, St James Leeds Hospital, High Royds Psychiatric Hospital, Dewsbury Hospital, Wycombe General Hospital, Great Ormond Street Hospital in 1971, Ashworth Hospital, Exeter Hospital, Royal Portsmouth Hospital, St Catherine’s Hospital in Birkenhead, and Saxondale Mental Health.
Health Secretary Jeremy Hunt said he wanted to be able to assure NHS patients that it would be “much, much harder” for abuse on such a scale to happen again by establishing whether NHS procedures were to blame.
He told BBC Radio 4’s World at One programme that the scale of the challenge for the NHS’s investigation into Savile’s abuse on its premises was “absolutely huge” because it would cover a period of about 40 years.
Labour shadow home secretary Yvette Cooper called for “a proper overarching review led by child protection experts into why everyone failed to stop Savile and what should be done now.”
“A myriad of small reviews and inquiries into how it could happen in different hospitals or the BBC are just not enough,” she said.
The Crown Prosecution Service also published a review of a decision in 2009 not to charge Savile with sexual offences in relation to four complaints made to police in Surrey and Sussex.
It found there was “nothing to suggest” the decisions were “consciously influenced by any improper motive on the part of either the police or prosecutors”.
However, it said further action might have been possible had “police and prosecutors taken a different approach”, adding the CPS prosecutor “did not probe… or seek to build a prosecution”.
If those coming forward had “been told that she was not the only woman to have complained, they would probably have been prepared to give evidence”.
Director of public prosecutions Keir Starmer apologised on behalf of the CPS. He said the report represented a “watershed moment” and pledged to enhance information sharing and provide prosecutors with additional training for the future.
The Department of Health said the Savile report showed the need to “learn lessons from his crimes” and the findings would feed into its own investigation.
Great Ormond Street Hospital called the report “extremely distressing,” adding the allegation relating to the hospital was “not reported at the time and therefore neither the police nor GOSH hold any records relating to the matter”.
Wheatfields Hospice said it was “appalled and dismayed” to hear the allegation against it and its “thoughts are with the individual involved and their family at this difficult time”.
Speaking on behalf of Broadmoor, West London Mental Health NHS Trust called the report “deeply distressing,” saying it was reviewing “thousands of files and records” to help with the police investigation.
Kim Harrison, a lawyer representing about 50 victims, said the report was one of the first steps on the way to victims getting “some kind of justice”.
But Kerry Griffiths, who was kissed by Savile when she was 13, told BBC News she did not feel other victims would “get any kind of closure” from the report.
BBC media correspondent Torin Douglas said the main target for victims making compensation claims would be the Savile estate. The BBC could also be targeted, he added.
Operation Yewtree has three strands – claims against Savile, claims against Savile and others, and claims against others. The report marks the end of the investigation into claims against Savile.
The allegations against Savile prompted a series of investigations, including three at the BBC and another by the Department of Health into his role at Broadmoor.
A report by former Sky News chief Nick Pollard into the the dropping of a BBC Newsnight investigation into Savile’s abuse said the decision was “flawed” and plunged the corporation into “chaos and confusion”.
Savile was a BBC Radio 1 DJ as well as a presenter of Top Of The Pops and Jim’ll Fix It on the BBC, and was knighted in 1990.