The incidents, which occurred over a two-week period, saw three young women followed into different blocks of flats in the west end of Glasgow and then attacked.
Strathclyde Police believe all three "frightening" incidents are linked and one man is responsible.
The first incident took place on Thursday January 24 at 4.20pm when a 23-year-old was followed along Great Western Road into a common close and indecently assaulted.
One week later on January 31 at 6.25pm, a 19-year-old woman was similarly attacked in the close of another block of flats on Arlington Street.
Three days afterwards, a 20-year-old woman was followed into flats in Kersland Street at 5pm on Sunday, February 3 before she was assaulted.
The suspect is white, aged 20-30, around 6ft to 6ft 2in, slim with black hair.
He was wearing a black top, possibly hooded, and black or dark trousers at the time of the assaults. Twice he was wearing a black beanie hat and once he had on a black skip cap.
Chief Inspector Alan Gibson said: "Each of these incidents were particularly frightening for the young women concerned. We are continuing inquiries in the local area, including viewing all available CCTV in an effort to identify the man responsible, and we have been deploying extra officers around the area to provide public reassurance.
"I would like to appeal to anyone who may have witnessed a man fitting the description of the suspect acting suspiciously around the areas and times of the incidents to get in touch with police. It is imperative that we trace this man."
Anyone with information is asked to contact police on 0141 535 3350 or anonymously via Crimestoppers on 0800 555 111.