4 July 2013 Last updated at 09:20 ET 
Madeleine was almost four years old when she disappeared from her parents' holiday apartment
Scotland Yard says it has "new evidence and new theories" in the Madeleine McCann case as it opens a formal investigation into her disappearance.
The Met Police said it still believed there was a chance Madeleine was alive and was investigating 38 "persons of interest" after reviewing the evidence.
Madeleine's parents, Gerry and Kate McCann, said the shift from review to investigation was "a big step forward".
Madeleine was almost four when she disappeared in Portugal in 2007.
Portuguese authorities dropped their investigation in 2008.
New witnesses Scotland Yard's review started in May 2011, after Prime Minister David Cameron responded to a plea from the McCanns, of Rothley, Leicestershire.
Det Ch Insp Andy Redwood, who is heading Operation Grange, said: "The review has given us new thinking, new theories, new evidence and new witnesses."
Continue reading the main story Analysis
Danny Shaw Home affairs correspondent, BBC News
A team of Scotland Yard's best detectives, assisted by police overseas, in an inquiry paid for by the Home Office, offers the best opportunity to find out what happened to Madeleine McCann. It may also be the last chance to do so.
Although the Yard can not put right mistakes made during the initial inquiry, when the Portuguese were accused of failing to seal off the area where Madeleine had been staying and gather all available forensic evidence, the new team has had access to all the files.
That has enabled detectives with no prior involvement to look at the case objectively and form fresh views about what may have happened to Madeleine. There is a real determination among officers to solve the mystery - and a quiet optimism that it may be possible.
His 37-strong police team is two-thirds of the way through examining 30,500 documents from files held by the Portuguese, private investigators and British police. Some fresh interviews have also taken place.
"We continue to believe that there is a possibility that Madeleine is alive," Det Ch Insp Redwood said.
"It is a positive step in our hunt for Madeleine that our understanding of the evidence has enabled us to shift from review to investigation."
Scotland Yard's decision to formally open its own investigation - which, like the review, will be funded by the Home Office - follows extensive discussions with UK prosecutors and the Portuguese.
BBC home affairs correspondent Danny Shaw said: "It's highly unusual - although not unprecedented - for a British police force to launch its own inquiry into an incident abroad."
In a statement, the McCanns said: "Kate and Gerry warmly welcome the shift in the Met's emphasis from review to investigation.
"It is clearly a big step forward in establishing what happened and, hopefully, towards bringing whoever is responsible for Madeleine's abduction to justice."
'Potential involvement' The Met said its review of tens of thousands of documents had "generated in excess of 3,800 actions", which in turn had "generated new findings and new witness evidence".
The 38 "persons of interest," who include 12 British nationals, are from five European countries - Portugal, the UK and three unnamed others.
Detectives say they need to find out more information and collect evidence on them and are not anticipating any immediate arrests.
The 12 UK nationals, who are not all currently in the UK, are believed to have been in Portugal at the time of Madeleine's disappearance.
The Metropolitan Police issued a computer-generated image (r) of how Madeleine might have looked aged nine
The Met said it was "at an advanced stage of dialogue" with the other countries involved, and enquiries would be continuing with their assistance in the months ahead "to establish more information about the individuals concerned and any potential involvement".
A foreign national resident abroad could not be prosecuted in the UK for any possible crime that may be linked to Madeleine's disappearance.
Detectives say Madeleine's parents, the friends the McCanns were with in Portugal and people known to the family before they went away are not suspects or people they need to investigate.
The investigation is currently closed in Portugal and as part of the country's criminal justice system it cannot be reopened unless judges are convinced there are solid grounds to do so.
British police have formally asked the Crown Prosecution Service to submit an International Letter of Request to Portuguese authorities for assistance in obtaining evidence relating to their inquiries.
The Met has asked for a small number of its officers to be present in Portugal for the inquiries there.
"Our working relationship with the Portuguese police is positive and now that we have moved to investigation we are requesting further specific assistance through normal judicial routes," Det Ch Insp Redwood said.
As part of the review, a computer-generated image of how Madeleine might look at the age of nine was created with the help of her family. They marked her 10th birthday on 12 May.
By May 2012 - one year into the review - the Home Office's costs had reached £1.9m. The Met Police said a more up-to-date figure for the cost of the review would be released in due course.
