Having started reviewing the case in 2011, Scotland Yard are still investigating, receiving £150,000 in new government funding to pursue the matter on 13 November 2018.
With interest in the case showing no signs of abating, the chances are that unless Madeleine is found, the theories will keep proliferating.
For now, though, the most commonly discussed scenarios seem to be:
The parents were involved
On September 7 2007, four months after Madeleine’s disappearance, Kate and Gerry McCann were made “arguidos”, formal suspects in the case.
The theory seems to have been that the two doctors killed their daughter by accident, possibly by giving her too much sedative in the hope that she would sleep while they went out to dine with friends at a tapas restaurant 50 yards away.
The McCanns are then supposed to have hidden the body, faked Madeleine’s abduction, and then, weeks later, with the search for the three-year-old in full cry, put the corpse in the boot of a hire car and hid it in some super-secret location.
The Portugese police appear to have been bolstered in the belief that this – or something like it – could be possible by the reactions of two sniffer dogs brought to the Algarve by British officers in July 2007.
The two dogs, one trained to detect human blood, the other the smell of dead bodies, were taken to numerous locations, but only gave alerts inside the McCanns’ holiday apartment. The cadaver dog later gave another alert signal while inside the Renault Scenic hire car that the McCanns had rented 24 days after their daughter went missing.
Analysis of hair and other fibres taken from the hire car and apartment were analysed by the UK’s Forensic Science Service. Four days after the FSS analysis was complete, the McCanns were made arguidos, with Portugese officials citing DNA evidence as one of the grounds for suspicion.
It later transpired, however, that neither the sniffer dog or DNA evidence were as watertight as some observers assumed.
Sniffer dog evidence has been the subject of research questioning its reliability, and in one US court case, a judge agreed with analysis of three cadaver dogs’ performances which found they were wrong 78 per cent, 71 per cent and 62 per cent of the time.
The DNA evidence, meanwhile, came with a major ‘health warning’ from the FSS.
An email from John Lowe, of the FSS team, stated that only 15 out of a set of 19 components of Madeleine’s DNA profile had been found in the hire car.
Mr Lowe cautioned: “The individual components in Madeleine’s profile are not unique to her; it is the specific combination of 19 components that makes her profile unique above all others. Elements of Madeleine’s profile are also present within the profiles of many of the scientists here in Birmingham, myself included.
“It’s important to stress that 50 per cent of Madeleine’s profile will be shared with each parent. It is not possible, in a mixture of more than two people, to determine or evaluate which specific DNA components pair with each other. … Therefore, we cannot answer the question: Is the match genuine, or is it a chance match.”
Kate and Gerry McCann had their formal arguido status removed in July 2008. They have always maintained their innocence. They have dismissed as “ludicrous” theories about how they might have used the hire car to dispose of the body in a location that Algarve locals didn’t know about.
They are also continuing their legal action against Goncalo Amaral, the former lead Portugese investigator in the case, who in 2008 wrote a book claiming they had faked their daughter’s abduction after her accidental death.
Burglary gone wrong
According to this theory, thieves broke into the apartment, then killed or abducted Madeleine when she woke up and saw what they were doing.
It is understood that the Scotland Yard review team has not yet ruled out such a scenario – even though you might expect it to involve thieves acting in panic, making hurried mistakes and leaving a trail of obvious clues in their wake.
At one time, though, the Met detectives had identified four Praia da Luz locals who seemed as if they might have fitted the bill. Their backgrounds, mobile phone contact and locations around the time of Madeleine’s disappearance all seemed to fit the pattern of men involved in a burglary that had gone wrong.
The men were questioned in 2014 – but Scotland Yard has now announced there was no evidence to implicate any of them and abandoned the case against all four.
Paulo Ribeiro, one of the men, has also told BBC Panorama how amazed he had been when approached by British detectives.
“I thought it was incredible,” he said. “I knew of nothing when the police arrived at my door with a piece of paper that had a drawing on it, saying it bore a likeness to me and that someone had said I was involved and that I looked like the person who had kidnapped Maddie.”
He and the other three men, were also backed by Pedro do Carmo, deputy director of the Policia Judiciaria, who told Panorama he had never considered them to be suspects.
He said: “I can only say that we questioned those people on request of the Metropolitan Police and only based on the request of the Metropolitan Police.
“We never questioned those people. We never saw or looked at those people as suspects of the crime.”
And, despite British detectives still considering the idea of a burglary gone wrong to be plausible, Carlos Anjos, the former head of the Policia Judiciaria officers’ union, told Panorama: “This burglary theory is absurd. Not even a wallet disappeared, no television disappeared, nothing else disappeared. A child disappeared.”
Abducted or killed by a local paedophile
In 2009 there were reports that some of those investigating Madeleine’s disappearance believed the Algarve had been “awash with paedophiles” when she went missing.
One source was quoted as saying: “There are 38 known sex offenders in the Algarve. The area is a magnet for paedophiles. There have been seven sexual assaults involving the children of tourists in the Algarve in the last four years.
“They all have the same modus operandi as Madeleine’s disappearance – that is, a break-in at a holiday apartment and children molested.
“Five happened before Madeleine’s abduction, and two afterwards. One took place a month before she vanished.”
Could a local serial paedophile have escalated his activities to the point where instead of leaving a molested foreign child in situ, they progressed to abduction or killing?
Names have been proposed, and either categorically denied or conclusively disproven.
In this respect, probably the most bizarre twist came last year when it emerged that the broadcaster, wit and ex-MP Sir Clement Freud, who had a Praia da Luz holiday home and who had befriended the McCanns, was also a paedophile who sexually abused girls for decades.
One of Freud’s victims told ITV News she felt “very uneasy” after discovering that the former politician had befriended the McCann family:
“He was a private man, not the type to suddenly befriend a couple who had intense media attention. It really jarred with me, left me feeling very uneasy. Nobody else would have thought Freud capable of abuse and rape but he did it to me.”
Some investigators, however, have cautioned against the whole idea of the involvement of a local paedophile. They point out that only very rarely do predatory paedophiles take the risk of sneaking or breaking into a building to get at a child.
Abducted by slave traders or paedophile child traffickers
There have been numerous variations of this theory, placing Madeleine in locations from Belgium to Africa.
In 2008, for example, it was reported that police were examining claims that Madeleine was taken on the orders of a Belgium-based paedophile ring that had placed an order for a “young girl”.
The highly organised gang, it was said, may even have taken a photograph of Madeleine beforehand so the Belgian paedophiles could confirm she fitted their requirements and give the go-ahead for the abduction.
Other theories have suggested Madeleine could have been taken to Lagos marina, five miles from Praia da Luz, and put on a boat bound for Morocco.
There have been reports of girls fitting Madeleine’s description being seen in Morocco around the time of her disappearance, and in the first few weeks of the hunt for their daughter, the McCanns did visit the north African country to appeal for information.
Morocco also fits into theories that Madeleine was sold into slavery: it is on the trafficking route to the Saharan desert country of Mauritania.
Mauritania was the last country in the world to abolish slavery, outlawing the practice only in 1981. There are rumours that significant numbers of people remain in slavery in the country to this day, and that gangs operating out of Mauritania sell children to rich Middle Eastern families.
Colin Sutton, an ex-Scotland Yard detective, told the Mirror last month: “The Mauritania line is certainly a possibility and needs to be looked at.
“If someone wanted to get a three-year-old child into Africa it’s the obvious route. The infrastructure and contacts for people smuggling are clearly there.”
Madeleine woke up, wandered from the apartment, and was involved in an accident
If Madeleine had woken up and gone to see her parents at the tapas restaurant, it has been claimed, there were a number of potentially fatal accidents that could have befallen her.
If she lost her way in the dark and took a wrong turn, a walk of less than 200 yards would have taken her to some roadworks.
It has been reported that workers repairing drains had dug a 6ft-deep, 4ft-wide trench. Had she fallen in, causing her to die or be knocked unconscious? And had she then not been noticed when the trench was filled in the next morning?
This theory has been disputed by the engineer and the foreman in charge of the works, who have both insisted the trench was checked by them and the police.
Other accident theories have posited a drink driver who runs over Madeleine, panics, and hides the body. A local might know hiding places where a body could lie undiscovered for ten years: the surrounding countryside is sparsely populated with plenty of scrubland and many old, long-abandoned wells.
Such scenarios, though, would require unusual behaviour from a little girl going in search of her parents.
Kate McCann has pointed out that Madeleine, coming up to her fourth birthday, would have had to have been able to open the curtains, slide open the patio door and then shut both of them behind her. Then she would have had to open and shut the garden gate leading to the road.
During her holiday, she had got to know the way that took her to the pool complex where her parents were eating. So why would this little girl have continued down the dark, lonely path leading to the roadworks, instead of turning towards the reassuring lights and noise coming from where she knew she could find her mum and dad?
This article was first published on 3 May 2017 by daily independent