To date 14,000 police man hours, thousands of knocked doors, hundreds of thousands of phone calls and $18 million in costs have failed to find the phantom killer who moves through Europe like a wraith and leaves dead innocents in her wake. Since that first murder she has killed twice and her DNA has been found at the site of a triple-execution that she may or may not have been involved in. Her genetic calling-card has also been found at the sites of numerous break-ins and petty robberies, upon a drug-filled syringe on a leafy pathway and in garden sheds in Austria.

Conventional thinking brands the woman without a face as a drug addict who kills and robs to feed her habit. But if that is so, police say she exercises an extraordinary restraint not usually found in junkies, whose craving normally causes them to implode, to make mistakes that throw them off their selfish, lethal quest. Last week German police resolved to step up the efforts of "Special Commission Parkplatz" - parking place, after the spot where a female police officer became one of her victims - in a bid to close the file on one of the new millennium's most baffling cases and to redeploy the 50 detectives now working on the case to other crimes. But good intentions cannot mask the fact that the phantom killer remains a mystery, and unless she is shopped by an accomplice or slips up, the chances are that she may well become the Jack the Ripper of her age: unknown and unaccountable for her terrible crimes.

One reason for the intensity of the hunt is that police want to avenge one of their own. She is the only suspect in the cold-blooded murder of Michele Kiesewetter, 22, a German policewoman who was killed in a car park at Heilbronn in April last year. Police believe the officer approached the woman and she panicked, killing her with a bullet in the face. Inside the BMW squad car: a DNA calling-card of "Frau Ohne Gesicht" ("woman without a face"). The DNA, smudged on the centre console and the armrest in the rear of the car, was fed into the Interpol database: it matched the 1993 tea cup killing and one in 2001 in the south-western German university city of Freiburg.

Joseph Walzenbach, 61, was a respected antiques dealer. He was found strangled with a length of garden twine, and a small amount of cash was taken from him - not more than the equivalent of $230. The DNA of his killer was found on him, on items in his small shop and smeared on the door handle. It was even smudged on the back of the "closed" sign which the killer placed in the window of the premises before fleeing. "After 2001 we had two murders - not enough to classify the perpetrator as a serial killer but with similarities: small amounts of cash stolen, the same modus operandi in the way the victims died, both killings committed indoors with no signs of a break-in. This in itself suggests the killer builds up a non-threatening rapport before being let in," Juergen Brauer, a state prosecutor who works closely with Special Commission Parkplatz, says. Juergen Bueller was seven as he walked along a leafy lane in Gerolstein, a spa town not far from the Belgian border, towards a playground. He felt something crunch under his foot and looked down to see glass, a needle and what looked like flecks of blood.

Now 15, he recalls: "I ran back home clutching this thing in my hand. I had no idea what it was; I had never seen a syringe before. The jabs they gave us at school were done with something that looked like a silver pocket torch. "I was quite excited and showed it to my mum, who went nuts. She took it off me and said: 'Now are you cut? Have you put it anywhere near your mouth? Did it go through your shoe into your foot?' Of course I didn't know anything about AIDS then, but I do now. I suppose I was lucky not to have cut myself. I just remember having a battery of tests in local hospitals that seemed to stretch on forever."

The syringe was found to contain traces of heroin and the DNA of the double killer. The German state computers and Interpol began cross-checking the DNA for other crimes. What appeared over the next few years was low-key in criminal terms but helped compile the portrait of a nomadic, wandering criminal moving at will across Germany and other countries. A fortnight after the discovery of the syringe a caravan was burgled on the outskirts of Mainz; the remains of a nibbled biscuit turned up the woman's DNA once more. On New Year's Day in 2003 at Dietzenbach near Frankfurt-am-Main an office was broken into and nothing but a coffee tin of loose change stolen. "It was a professional job," Guenter Horn, a high-profile prosecutor liaising with police, says. "She left no fingerprints. But she did leave a scraping of skin, and that was enough to pin the job on the strangler." So it went on. December 2003, a car stolen in Heilbronn with her genetic fingerprints on the petrol cap. In autumn 2004 the woman went to the Austrian Tyrol. She broke into garden sheds along the road towards Innsbruck, discarding a pair of tracksuit bottoms, a hooded cardigan and other items. The killer's DNA also turned up at burglary scenes in France.

September 2005: a bar robbery in Karlsruehe and the DNA found on beer bottles and a wine glass. August 2006 in Besancon, France: DNA on a toy pistol used in a hold-up for $170 from the till of a dry-cleaners. In all, there have been about 30 break-ins and hold-ups as well as the murders that have yielded her DNA identity. In May 2005, in the city of Worms, a Gypsy turned a gun on his brother. Police found the phantom's DNA on one of the bullets.

Knowing her dress style, drug use and cold-bloodedness, police went on television in April 2005 with an appeal to the public for tips, but to no avail. Then came the event in April last year that sent shock waves through German society. Michele Kiesewetter, 22, and her colleague were eating rolls and drinking coffee during a lunch break in their patrol car. They had been assigned to an undercover drugs squad in Heilbronn. At least two people climbed into the back of the car and shot both in the head. Kiesewetter died instantly; her partner lingered in a coma for months before the bullet lodged behind his right eye was removed. He remembered nothing from the incident. Nothing was taken from them except their handcuffs.

"It was brutal, apparently random and with no apparent motive," Chief Superintendent Horst Haug of Special Commission Parkplatz says. "What are we dealing with here? And who is the accomplice?" Who indeed. Police revealed that other DNA traces were found at crime scenes indicating she sometimes operated in tandem. But no two crime scenes yielded the same DNA, indicating she picks up and discards helpers with the same casual abandon with which she kills.

In May this year, at Saarhlzbach in western Germany, near the Luxembourg border, a woman at a fishing lodge was knocked unconscious and $600 was stolen. Left behind, again, was the same DNA and that of an unknown sidekick. Kurt Kletzer, a noted Viennese psychiatrist who recently drew up a profile of the Austrian incest father Josef Fritzl for a book I wrote called Monster about his life and crimes, says the woman without a face was "intriguing and disturbing" in equal measure. "Judging by the facts alone, we are dealing with an ordered personality that maintains composure even though the crimes and the evidence, such as the syringe, indicates drug dependence. Stealing small amounts of cash is an indicator that she has needs which must be served, but she doesn't go for the 'big score' which might trap her. The fact that she sometimes uses an accomplice suggests she is a manipulator and an adept one at that.

"Like Fritzl, she is able to project an aura of normalcy while being anything but. She is compelled to murder to feed her own habit, thus reducing the victim to the status of a worthless object. These psychopathic traits would have been formed at a very early age and I would venture that the police are looking for someone from a damaged home life, perhaps a foster child or orphan, a child who was abused or whose carers were addicts themselves. "What doesn't fit at all is the murder of the policewoman and the attempted killing of her partner. Unless they thought that they would have drugs, the act may be seen as the ultimate lashing out at authority."

As Special Commission Parkplatz cranks up its probe, an intriguing court case is under way. This year police found cells of the suspect's skin after they stripped and analysed all the upholstery, carpets and lint from the car of the man now on trial for a triple murder. The man, a former paid police informant, is suspected with another of killing three Georgian dealers who had come to Germany to buy second-hand cars. Their bodies were dumped in the River Rhine at the end of January. The mysterious murder was "an act of religious hatred", one of the two accused claimed on Monday at the start of the trial. Testifying in Arabic, the Iraqi-born defendant, 40, declared his innocence and accused the other defendant, a 26-year-old Somali, of murdering the three as an expression of "a special form of fundamentalism".

Police said they had employed the Arabic-speaker as a spy. He told the court his job was to observe the Somali, who had found the victims' Christianity objectionable. Prosecutors in the town of Frankenthal reject that claim and say the two accused killed the Georgians simply to seize their wads of cash. The dealers had come to Germany to obtain second-hand cars to take home.

The Somali, speaking fluent German, also insisted on his innocence, saying he had been present but was surprised by the brutal murders and dared not object for fear that he too would be killed. But both deny knowledge of the woman without a face, whose DNA was found in the car where the men were executed. Could she be the killer? Or a participant in it? Both men say they are at a loss to explain how her double helix came to be in the car. Police are compiling a thorough history of the car: who owned it and where it has been seen.

"We are as baffled as everyone else about how her DNA came to be there," said Haug. "But it is one more piece in the mosaic. All criminals, no matter how deadly or how clever, all slip up eventually. We just hope we catch her before she kills again." Alan Hall is an Australian journalist living in Berlin.