German Chancellor Angela Merkel has voiced outrage over the sexual assaults and thefts that male gangs inflicted on women in Cologne on New Year's Eve.
Women have made at least 90 criminal complaints to police about the harassment by gangs at Cologne's main railway station on Thursday night.
Germans have been shocked by the scale of the attacks, involving many groups of drunk and aggressive young men.
Witnesses and police said the men were of Arab or North African appearance.
Mrs Merkel called Cologne Mayor Henriette Reker on Tuesday and expressed her "outrage over these disgusting attacks and sexual assaults".
The chancellor said everything must be done "to find the perpetrators as quickly and comprehensively as possible and punish them, regardless of their origin or background".
Up to 300 women demonstrated against the violence near the scene of the attacks on Tuesday evening. One placard read: "Mrs Merkel! Where are you? What do you say? This alarms us!"
There is an intense debate in Germany about refugees and migrants, who arrived in record numbers last year. But Mayor Reker urged people not to jump to conclusions about the Cologne assailants.
"It's completely improper... to link a group that appeared to come from North Africa with the refugees," she said, after crisis talks with the police.
Cologne police chief Wolfgang Albers called it "a completely new dimension of crime" and rejected criticism of his force's handling on the night.
Women were also targeted in Hamburg and Stuttgart in similar attacks, but on a smaller scale.
In Cologne at least one woman was raped and many were groped, including a volunteer policewoman.
One man described how his partner and 15-year-old daughter were surrounded by an enormous crowd outside the station and he was unable to help. "The attackers grabbed her and my partner's breasts and groped them between their legs."
Most of the crimes reported to police were robberies.
A policeman told the city's Express news website that he had detained eight suspects. "They were all asylum seekers, carrying copies of their residence certificates," he said.
However, there was no official confirmation that asylum seekers had been involved in the violence. Germany' migrant influx triggered marches by the anti-Islam Pegida movement last year.
Police were deployed outside the station because of the crowds on New Year's Eve, but failed to spot the many attacks, according to reports.
Germany's New Year shock - by Damien McGuinness, BBC News Berlin
The pretty Christmas market and medieval setting may look idyllic, but at Christmas and New Year the area around Cologne Cathedral is a notorious danger zone when it comes to pickpockets and theft.
Now the sexual harassment, and in one case rape, of dozens of women has shocked Germany.
What is particularly disturbing is that the attacks appear to have been organised. Around 1,000 young men arrived in large groups, seemingly with the specific intention of carrying out attacks on women.
Police in Hamburg are now reporting similar incidents on New Year's Eve in the party area of St Pauli. One politician says this is just the tip of the iceberg.
And there are real concerns about what will happen in February when the drunken street-parties of carnival season kick off.
Cologne will stage carnival events in February, with hundreds of thousands of revellers expected in the streets, as on New Year's Eve.
The police chief said "the assailants' behaviour is a real concern for me, also because of the carnival".
After the crisis meeting he said that in future more police would be deployed at big city events and extra surveillance cameras would be installed.
Mayor Reker promised preventive measures ahead of the carnival. The authorities would warn young women about possible risks and "give a better explanation to people from other cultures about the meaning of carnival", she said.
"We need to prevent confusion here, about what constitutes happy behaviour and what is utterly separate from openness, especially in sexual behaviour," she said.
She called the attacks "monstrous", saying "we cannot allow this to become a lawless area".
A British woman visiting Cologne said fireworks had been thrown at her group by men who spoke neither German nor English. "They were trying to hug us, kiss us. One man stole my friend's bag," she told the BBC.
"Another tried to get us into his 'private taxi'. I've been in scary and even life-threatening situations and I've never experienced anything like that."
Cologne news website Koelner Stadt-Anzeiger says the suspects were already known to police because of frequent pickpocketing in and around the central station.