The disgraced TV presenter used his celebrity to "hide in plain sight", said the report, Giving Victims A Voice, by Metropolitan police and children's protection group, the NSPCC.
Evidence of 214 criminal offences has been recorded against Savile's name in 28 police forces, including 34 rapes.
Five offences are now known to have been committed in Scotland, with Savile claiming victims across Strathclyde, Fife, Grampian, Lothian and Borders and in the Northern Constabulary area, in which he owned a holiday cottage at Glencoe.
Police confirmed that Savile had allegedly abused one victim in the Highlands, but the offence had not been committed at the Glencoe cottage. The victim now lives in the Strathclyde area.
Further victims now living in Scotland have reported offences that occurred in England.
Home Secretary Theresa May has been called on by her Labour Shadow, Yvette Cooper, to order an over-arching review into the scandal to replace the separate inquiries currently planned.
The report discloses that Savile sexually abused a dying teenager at a hospice, one of 14 medical sites where he used to prey on his victims.
Commander Peter Spindler, who led the investigation, said: "We hope this report gives some comfort to his hundreds of victims. He spent every minute of every waking day thinking about it [abusing children] and whenever the opportunity came along he took it."
He added that "cunning" Savile's campaign of sex crimes were "vast, predatory and opportunistic", and he had effectively "groomed the nation".
The disgraced broadcaster, who was awarded a knighthood and enjoyed public affection for his charity work which gave him access to vulnerable children, abused patients at Stoke Mandeville Hospital and pupils at Duncroft School.
He committed 14 offences at schools, sometimes after children had written to him asking to appear on his Jim'll Fix It.
While the bulk of his offending was committed between 1966 and 1976, when he was between 40 and 50, the report said, it is believed he also molested a young teenage girl at the last Top of the Pops recording in 2006, when he was 78.
The joint report between the Metropolitan Police and the NSPCC stopped short of pinning any blame on other institutions that may have "missed past opportunities" to stop Savile.
In a separate report, Keir Starmer, director of public prosecutions in England and Wales, made a public apology after admitting Savile could have been charged for offences against at least three victims before his death in 2011.
Glasgow woman Caroline Moore, who said Savile abused her as she recovered in the children's ward at Stoke Mandeville Hospital in 1971, said yesterday: "I find it quite scary that people who had the power to stop him didn't use that power and he went on to do the most horrific things to people in the most awful circumstances."
Kevin Cook said he was abused by Savile and a second man when he visited BBC studios for Jim'll Fix It. He said: "I'm shocked by the time it's gone on and the amount that's gone on."
A total of 450 people have come forward since last October alleging sexual abuse – of whom 73% were children at the time of the offence. From October 3 to November 21 last year, 233 referrals to police from an NSPCC helpline were generated specifically in relation to Savile, with 3.4% of the referrals from calls in Scotland.
Matt Forde, head of NSPCC Scotland services, said: "It is simply horrific Savile was able to sexually abuse children across the UK over six decades, including different Scottish regions.
"Sexual abuse has not died with Savile, nor is it primarily about celebrities. The number of children in Scotland on the child protection register because of sexual abuse has increased every year since 2008 and we know from research that this is just the tip of the iceberg."
It's scary that the people who had the power to stop him didn't use it and he went on to do the most horrific things