Detectives have appealed for information about what they said was a "targeted" and pre-planned attack in the centre of Glasgow.
Two men - both aged 33 - were injured with what is thought to have been a knife before the attacker, also 33, inflicted fatal injuries on himself.
The incident has been referred to the Police Investigations and Review Commissioner (PIRC).
One of the injured men remains in hospital in a "stable" condition.
Witnesses had told of seeing a man with a knife at the scene, and pools of blood on the road.
Police were called to West Nile Street at about 16:00 on Thursday after the first man was assaulted.
The attacker ran off before attacking a Glasgow City Council community warden who had tried to intervene.
The suspect then started to injure himself. When police officers arrived, they used a spray on the man and disarmed him.
He was then taken to Glasgow Royal Infirmary but died a short time later.
The community warden was treated for an injury to his arm and then released.
Det Supt Laura Thomson said: "From our inquiries so far, we know that the initial attack was targeted - it was not random; it was planned and we are not looking for anyone else in connection with it.
"However, although we have had a great response from members of the public who were there at the time, we need people to come forward with any information that may assist the investigation.
"Officers, including specialist forensic teams, are still at the scene carrying out inquiries and checking CCTV. Additional officers are in the city centre to provide public reassurance."
A spokesman for PIRC said: "The PIRC is undertaking an investigation into the circumstances leading up to the death in custody of a 33-year-old man in Glasgow on Thursday 13 April 2017.
"The man was involved in a prior incident in West Nile Street before he inflicted serious injuries upon himself.
"Captor Pava - an incapacitant spray - was then discharged by police and the man was detained.
"He was then taken by ambulance to Glasgow Royal Infirmary for treatment but died a short time later."