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 investigation, said Savile had "groomed the nation".
And the NSPCC said Savile was one of the most prolific sex offenders in its 129-year history.
The former BBC presenter of Top Of The Pops and Jim'll Fix It, who also worked as a Radio 1 DJ and received a knighthood in 1990, 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.
NSPCC director of child protection advice and awareness Peter Watt said: "The sheer scale of Savile's abuse over six decades simply beggars belief.
"He is without doubt one of the most prolific sex offenders we have ever come across and every number represents a victim that will never get justice now he is dead."
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 1977In 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".
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% 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 1965-95 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.
Jimmy Savile inquiriesHealth 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.
Operation Yewtree Scotland Yard criminal investigation into sexual abuse claims against Savile and others linked to the presenter
BBC investigation led by former Sky News head Nick Pollard into management failures over the dropping of Newsnight report about Savile
BBC investigation led by former Appeal Court judge Dame Janet Smith into corporation's culture and practices during Savile's career and current child protection and whistle-blowing policies
BBC investigation led by Dinah Rose QC into handling of past sexual harassment claims
Department of Health investigation into its own conduct in appointing Savile to lead a "taskforce" overseeing management of high security psychiatric hospital Broadmoor in 1988
Director of Public Prosecutions review into decisions by the Crown Prosecution Service not to prosecute Savile in 2009
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.
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 said further action might have been possible had "police and prosecutors taken a different approach", prompting director of public prosecutions Keir Starmer to apologise on behalf of the CPS and say the report represented a "watershed moment".
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".
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.
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.
The Dame Janet Smith Review, which is looking into the culture and practices of the BBC while Savile was an employee, said it had now been contacted by more than 350 people. The investigation is ongoing.