The bodies of Helen Scott and Christine Eadie were found in fields about six miles apart in East Lothian in October 1977.
Both of the 17-year-old girls were naked from the waist down, and had been horrifically beaten and sexually assaulted.
Two years ago, convicted killer Angus Sinclair, 64, was cleared of committing what had become known as the World's End murders after his trial collapsed.
The case led to the Scottish government ordering a review of the country's double jeopardy rule, which prevents someone standing trial twice for the same offence.
BBC Scotland investigations correspondent Mark Daly has spent the past four months examining whether Sinclair could be a serial killer responsible for murdering at least seven women.
Even before his controversial acquittal two years ago for the 1977 World's End pub murders, Angus Robertson Sinclair was already one of the most dangerous sex criminals Scotland had ever seen.
During my four month investigation into Sinclair's full and brutal criminal past, I would learn how he managed to stay under the police radar for so long.
I would meet the FBI criminal profiler who told me about the evidence linking Sinclair to at least other four murders.
I was able to speak to the detectives who had pursued him down the years, and also, more tellingly, I met some of his victims who had lived to tell the tale.
All of this painted a chilling picture of a Jekyll and Hyde character, whose brutally violent streak and obsession with sex would wreak havoc across central Scotland during the 1970s and early 80s.
I uncovered a psychiatrist's report about Sinclair. It said: "I do not think that any form of psychotherapy is likely to benefit his condition and he will constitute a danger from now onwards. "He is obsessed by sex, and given the minimum of opportunity, he will repeat these offences, irrespective of what promises he may give to the contrary."
This was written when Sinclair was just a boy, after he committed his first murder in 1961, aged just 16. And its chilling warnings would not be heeded.
He had lured his eight-year old neighbour Catherine Reehill and raped and strangled her.
The calculated manner in which he disposed of the body and tried to cover his tracks shocked police.
He even called the ambulance himself, telling the operator, that "a wee girl has fallen down the stairs".
But detectives couldn't break him down, so they turned to his older brother John, who persuaded him to confess.
John, who has never spoken publicly about his brother before, told me he wishes he had stopped Angus in his tracks there and then.
He said: "I would have done time for him. I would have killed him. If I'd known years ago I'd have pushed him in the bloody canal. And all these people, all these girls would never have had that."
John told me he cut his brother off forever from that day onwards.
He said: "I would just have nothing to do with him, nothing do to with him. He can rot in prison."
But Angus Sinclair's life of crime was only just beginning at that point. He served just seven years for that killing.
In 1977 there was a spate of murders across Scotland. Six young women in seven months disappeared after nights out, and were found dumped on deserted farmland or waste ground.
Two of those victims were World End teenagers Helen Scott and Christine Eadie. They were discovered a few miles apart and had been bound, gagged and strangled in almost identical circumstances.
The biggest manhunt in Scottish police history was launched. But it would lead nowhere.
Police did not link the killings to the other four murders which had happened within a few months of each other.
Frances Barker, 37, Anna Kenny, 20, Hilda MacAulay, 36, and Agnes Cooney, 23, had all been killed and dumped in strikingly similar circumstances.
In the meantime, between 1978 and 1982, Angus Sinclair would rape or indecently assault countless children across Glasgow.
I tracked down two of his victims. One of them, Jean, now a grown woman, is breaking her silence for the first about her ordeal at the hands of Angus Sinclair. She told me how he had lured her into a tenement.
She said: "He asked me if I could do him a favour and go and take his mum's change up to her and he told me what close and what door."
Jean then described how she saw him coming up the stairs and tried to run.
She said: "I told him I need to go and then that's when he just grabbed me and he said listen I've got a knife and if you don't do as you're told, then I'll kill you.
"I thought I'll just do as I'm told and be quiet. So I just shut my eyes and just hoped he'd go away."
It is not known how many children Sinclair attacked during this period. He was finally caught in 1982, and pleaded guilty to 11 charges of rape and indecent assault, although he admitted his victims could have numbered in the hundreds. He was sentenced to life.
Almost 20 years later, when he was being prepared for parole, he was given his second life sentence after a cold case review of another murder: the 1978 killing of Mary Gallacher.
DCI Brian Murphy of Strathclyde Police, who led the investigation into the Gallagher case, said Sinclair had shown no remorse.
He told me: "He's one of the most evil people I've certainly interviewed, in my life. There were things going on in his head which clearly I couldn't reach."
But three years later in 2004, some ingenious detective work and cutting edge science was about to put Sinclair in the frame for World's End.
His DNA, and that of his dead brother-in-law, Gordon Hamilton, was found on a semen stain taken from Helen Scott's coat.
And Operation Trinity, the joint Lothian and Borders/Strathclyde Police investigation, was also looking at the unsolved Glasgow murders, and for the first time was treating all six as connected.
They had built a strong circumstantial case around the killer's MO, as well as the DNA found at World's End.
They had reviewed every murder in Scotland since 1968 - more than 1,000 - and found that all six murders bore uniquely similar characteristics.
Scottish police decided to ask for a second opinion, and turned to the world's foremost authority on serial killers, the FBI.
Special agent Mark Safarik spent four months poring over the evidence. Speaking for the first time about the work he carried out for Operation Trinity, he told me he was confident all six murders had been the work of a lone serial killer.
He said: "They brought six cases, and really the request for us was, can you look at all six of these cases and render an opinion about whether you think these cases are linked or not? And we requested all the case information, photographs, autopsy reports, anything that they had back from 1977.
"And after our analysis it was really clear to me, I felt very strongly, that all these crimes had been committed by the same offender."
Scottish police planned to charge Sinclair with all six murders. But there were two problems:
Strathclyde police had lost all the evidence from the unsolved Glasgow murders.
But even more embarrassing than that, was that someone was already in prison for the murder of Francis Barker - the first in the sequence from 1977 - and had been for almost 30 years.
The crown decided to drop all four Glasgow cases, and go all out on World's End.
But a case that had taken almost 30 years to get to court was over in just two weeks after the judge ruled there was no case to answer.
Prosecutor Alan McKay failed to lead some potentially crucial DNA evidence that suggested Sinclair had tied the knots on the ligatures used to bind and strangle the girls.
Former Deputy Chief Constable Tom Wood said he was shocked by Mr Mackay's decision.
He said: "I still don't understand why that supporting evidence was not led.
"And I mean, while I can't speak for every police officer in Scotland, I certainly know that for those who were on the investigation with me and who are still serving officers and can't speak openly about this, there is still an amazement that it wasn't done."
I asked Mr Wood if he was certain that Angus Sinclair was guilty of the World's End murders.
"I have no doubt whatsoever", he said.
After the collapse of the trial, amidst bitter recriminations, the Lord Advocate Elish Angiolini took the unprecedented step of addressing the Scottish Parliament about the crown's failure to secure a conviction.
The law of Double Jeopardy, which means that Sinclair cannot be tried for the same crime again, is currently under review.
He is still in prison for the Gallacher murder, and is eligible for parole in 2016.
But the BBC understands the Crown has carefully preserved all of the trial exhibits, and is relooking at all aspects of the case - particularly the DNA - with a view to launching a fresh prosecution if the law changes.
Morain Scott, the father of World's End victim Helen Scott, is watching developments closely. His search for justice has taken a heavy toll.
He said: "I still think he should be brought to justice on the case or any other cases if there are others, to give peace of mind to the victims, because that's what we are. We're victims of a wrongdoing.
"But you've got to carry on, because that's what I'm striving for, justice. Nothing else, just justice for the girls."
Scotland's Secret Serial Killer is available on the BBC iPlayer.