Two 19-year-olds, a man and a woman who were university students, were found dead on a city centre street with stab wounds after police were alerted about 4am (0300 GMT) on Tuesday.
Another 50-year-old man was also found dead with knife wounds on a road about three kilometres away.
A van, stolen from the 50-year-old victim, was then driven at three people, leaving one critically injured in hospital.
After the vehicle was stopped, police used a stun gun to arrest a 31-year-old man, and say they are not looking for any other suspects.
Counter-terrorism officers are helping with the investigation but Nottinghamshire Police's Chief Constable Kate Meynell said they were keeping an open mind as to what happened.
The BBC reported the suspect was believed to be a migrant of West African origin with a history of mental health issues.
"We are still in the early stages of the investigation and need to determine the motives behind these attacks," Meynell said.
The incident has shocked the city, particularly the student community, with Nottingham home to two universities with more than 50,000 students.
British media said one of the two teenage victims, named as Grace Kumar, had played hockey for England's Under 18 team.
The other, Barnaby Webber, was said to be a keen cricket player.
The University of Nottingham students were attacked as they returned home from a post-exam party, The Times reported.
Webber's parents said their son was "a beautiful, brilliant, bright young man, with everything in life to look forward to".
"Complete devastation is not enough to describe our pain and loss at the senseless murder of our son," they said in a statement released to media.
