Police were called to the victim's flat in Dobbie's Loan Place near Glasgow Caledonian University at around 11.30pm on Wednesday after friends became concerned that she had not been seen for three weeks.
Officers discovered the woman's body inside the property and last night confirmed that they were treating her death as murder, saying they believed she had been the victim of an unprovoked attack.
A post-mortem examination was carried out yesterday to determine the cause of death, and the police are appealing for help to establish her activities in the run-up to her murder.
Police have not yet formally identified the victim, although officers have been carrying out door-to-door inquiries with neighbours in the area.
They are also in the process of gathering CCTV footage and reviewing images in an attempt to trace her last movements and identify possible suspects.
The cause of death has not been made public.
Police inquiries so far have revealed that the victim was last seen on Wednesday 9 October, when she attended at Woodside Health Centre in Barr Street, in the Maryhill area of Glasgow.
Detective Chief Inspector Colin Carey, from Police Scotland's Major Investigation Team, said: "We are trying to establish a motive for this attack as it appears to have been unprovoked.
"We know that the woman was seen on Wednesday 9 October 2013 and we need to establish her movements from then on, who she saw, where she went, so if anyone has any information please do get in touch with us."
Police Scotland said it would be making additional patrols available in the area surrounding the crime scene.
Anyone who has any information about the incident is urged to speak to officers on patrol in the area, or contact detectives at Maryhill police office or Crimestoppers anonymously.