The Dame Janet Smith review identified a total of 72 victims, including eight who were raped.
Her report said BBC staff had known of the complaints about Savile but did not inform senior management.
BBC Trust chairman Rona Fairhead said the corporation had failed the victims.
Dame Janet's review found many, but not all, of the most serious incidents of rape, attempted rape and sexual assault were carried out by Savile at his own premises and not at the BBC.
But she said Savile would also "gratify himself sexually" on BBC premises whenever the opportunity arose, and that she had heard of incidents which had taken place at the corporation's Glasgow studios as well as in cities including London, Manchester and Leeds.
Dame Janet said Savile, who had a cottage in Glen Coe before he died at the age of 84 in 2011, and another former BBC presenter, Stuart Hall, were "serial sexual predators" and the BBC missed five opportunities to stop their misconduct.
She detailed an incident involving a 12-year-old girl, who travelled from Scotland to visit her aunt in London in December 1973.
According to the report, the girl persuaded her aunt to take her to see a recording of Top of the Pops, which Savile was presenting while dressed in a Womble outfit.
It said she became separated from her aunt inside the studio, and after the show finished she and a 10-year-old boy were asked by a man whether they would like to meet Savile.
They were taken to his dressing room, where they were offered fizzy drinks and biscuits by the man, before Savile came in, still wearing his Womble suit but without the head.
After chatting with the two children, the girl watched as Savile raped the boy, who was saying "don't, don't".
Savile then told the children "It's OK. It's our special secret", before sexually assaulting the girl.
The report said the girl later walked to the tube station, and told her aunt that she had seen Slade and had had pop and biscuits with Savile himself.
It added: "She did not tell anyone what had happened. She thought it was her fault and that she might go to hell for what she had done.
"Of course, she was a child in a vulnerable situation and what happened to her was not in any way her fault."
In April 2014, the victim saw a newspaper advertisement placed by a firm of solicitors asking for anyone who had been attacked by Savile to come forward.
She contacted them, and it emerged that the boy who was raped while she watched had also contacted the legal firm.
In his lifetime, millions knew Jimmy Savile as an eccentric TV personality.
He was one of Britain's biggest stars, a larger-than-life character who was known for tea time TV favourites such as Top of the Pops and Jim'll Fix It, as well as stints on BBC Radio 1.
He was also, to some, Saint Jimmy, a diligent fundraiser who raised Â£40m for charity.
But, a year after his death in 2011, allegations of abuse surfaced.
It transpired that he was, in fact, one of the UK's most prolific sexual predators.
He had been exploiting his status to prey on hundreds of people - girls and boys, men and women, but mostly vulnerable young females.
Profile: The Jekyll and Hyde TV presenter
Dame Janet's report also detailed an account of Savile going into his camper-van with a young woman while he was in Scotland to do a long-distance charity run.
But the witness told the inquiry: "I have no idea what took place when they went into the van, for all I know she was getting an autograph, a photograph or they were chatting or he was doing an interview with her.
"I do not know if his team knew any more than I. She was only in the van for a short time. A short time later i.e. within the hour maximum that I was present, an old lady brought Savile a pot of marmalade and said that it was for the work that he did for young people".
Tony Hall, director general of the BBC, apologised to the victims and said: "The BBC failed you when it should have protected you.
"A serial rapist and a predatory sexual abuser both hid in plain sight at the BBC for decades.
"What this terrible episode teaches us is that fame is power, a very strong form of power and like any form of power it must be held to account... and it wasn't."
Jimmy Savile abused some of his victims at the BBC's studios in Glasgow and also sexually assaulted a Scottish girl who had travelled to London to watch Top of the Pops, a report has found.