Witnesses described a motorist deliberately driving into families walking home from school.
South Wales Police said a 32-year-old woman was killed. Other victims, including six girls aged 23 months to 12-years-old and a nine-year-old boy, were taken to hospital for treatment.
An unidentified 31-year-old man has been arrested and a van has been seized. Police also say they are seeking witnesses who saw a black Renault Clio, which may be linked to the man they are questioning.
Authorities said that there had been at least five separate hit-and-run incidents across the city.
Det. Supt. Paul Hurley told a news conference the suspect left his van and assaulted people between the crashes. He said there was also a weapon involved but that it wasn't a firearm.
A woman who works in a store said a shopper in her mid 20s and two young girls, aged about 10, had bought some coffee before leaving her shop.
"All of a sudden I heard a bang. I looked out and across the road, one girl was on the floor and the other girl was screaming," Shady Taha, 29, told the BBC.
"I heard a [vehicle] speed off but I did not see it."
Ramesh Patel, who runs a store close to the site where the woman was killed, said customers had described how the van driver appeared to target pedestrians.
"Customers have said a white van purposely drove at people," Patel said.
Diane Quick told BBC news that her niece and her great-niece Amelia, aged two, were both injured.
"They were hit from behind by the van. Amelia was in the pram [and] she has bruising." Amelia's mother has a broken leg.
The incidents took place as parents were collecting children from nearby schools and walking home.