Hundreds of people have protested in the German city of Cologne over sexual assaults and thefts carried out by groups of men on New Year's Eve.
Some held up signs demanding action from Chancellor Angela Merkel.
Mrs Merkel has expressed outrage over the "disgusting attacks" and said everything must be done to find those responsible.
Witnesses and police said that the men involved were of Arab or North African appearance.
Political leaders have warned Germans not to link the violence to the influx of more than one million refugees and migrants in the past year.
"It's completely improper... to link a group that appeared to come from North Africa with the refugees," Cologne Mayor Henriette Reker said, following talks with police.
Interior Minister Thomas de Maiziere agreed there should not be any general suspicion towards refugees at least "at this stage of the investigation".
"But if North Africans were the perpetrators, for which there is some indication, there should not be a taboo and people should not gloss over it."
The scale of the attacks, involving groups of drunk and aggressive young men, has shocked the country.
There is an intense debate in Germany about refugees and migrants who arrived in record numbers last year, many of them fleeing the conflict in Syria.
Women have made at least 90 criminal complaints to police about harassment by gangs at Cologne's main railway station on Thursday night.
At least one woman in Cologne was reportedly raped and many were groped, including a volunteer policewoman.
Women were also targeted in Hamburg and Stuttgart in similar attacks, but on a smaller scale.
Up to 300 people, mostly women, demonstrated against the violence near the scene of the attacks on Tuesday evening.
Some held placards reading: "Mrs Merkel! Where are you? What do you say? This alarms us!"
Police were pictured stopping and questioning men near Cologne's central station on Tuesday.
However, the city's police chief, Wolfgang Albers, said no arrests had yet been made over the New Year's Eve attacks.
"We don't currently have any suspects, so we don't know who the perpetrators were," he said.
"All we know is that the police at the scene perceived that it was mostly young men aged 18 to 35 from the Arab or North African region."
He called it "a completely new dimension of crime" and rejected criticism of his force's handling on the night.
Justice Minister Heiko Maas warned against using the attacks to bolster anti-refugee sentiment.
"In criminal law what's important is proving a crime, and everyone is equal before the law," he said.
"It doesn't matter where someone comes from, it matters what they did and that we can prove it."
Mrs Merkel called Ms Reker on Tuesday to discuss the attacks.
She said everything must be done "to find the perpetrators as quickly and comprehensively as possible and punish them, regardless of their origin or background".
The mayor promised preventive measures ahead of Cologne's carnival in February, when hundreds of thousands of revellers are expected on the streets.
One man described how his partner and 15-year-old daughter were surrounded by a 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 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."