Image copyright Google Image caption The Tree of Life Congregation Synagogue in Pittsburgh
Reports say he barricaded himself in a room at the synagogue when police approached.
Emergency services arrived at the building at about 10:00 local time (14:00 GMT), and gunshots could be heard.
Pittsburgh’s Public Safety Director Wendell Hissrich later confirmed Mr Bowers was in police custody and was being treated in hospital.
The crime scene was “horrific”, he told reporters. “One of the worst I’ve seen, and I’ve [worked] on some plane crashes. It’s very bad,” he added.
He said that two officers were injured in an “initial confrontation” and that a further two Swat officers were later hurt by the gunman when they entered the building. He said that no children were among the casualties.
Image copyright Reuters Image caption Officers cordon off the area outside the synagogue
What do we know about the gunman?
US media said he had shouted “All Jews must die” as he carried out the attack.
Social media posts by someone with the name Robert Bowers were also reported to be full of anti-Semitic comments.
Pittsburgh FBI’s special agent in charge of the investigation, Bob Jones, told a press conference that he did not know if Mr Bowers was known to authorities prior to events on Saturday.
He said that any motive remains unknown but that authorities believe he was acting alone.
Mr Jones added that the investigation was “in the early stages”. “We will look at every aspect of the suspect’s life,” he said.
Mr Bowers is receiving treatment for what has been described as multiple gunshot wounds.
What has been President Trump’s reaction?
He called the shooting a “terrible, terrible thing”.
“To see this happen again and again, for so many years, it’s just a shame,” he told reporters on Saturday.
He described the gunman as a “maniac” and suggested the US should “stiffen up our laws of the death penalty”.
“These people should pay the ultimate price. This has to stop,” he said.
Gun control around the world Fight or flight: Would you tackle a gunman?
Mr Trump added that the incident had “little to do” with US gun laws. “If they had protection inside, maybe it could have been a different situation,” he said.
The president later appeared at the Future Farmers of America Convention in Indianapolis, saying: “There must be no tolerance for anti-Semitism. It must be condemned and confronted everywhere and anywhere it appears.”
Other world leaders to condemn the attack include Israeli Prime Minister Benjamin Netanyahu, who said he was “heartbroken and appalled”.
“We stand together with the American people in the face of this horrendous anti-Semitic brutality,” he said in a video message.
Canada’s Prime Minister Justin Trudeau said his country’s hearts “are with the Jewish community in Pittsburgh today”.
German Chancellor Angela Merkel said: “We all have to stand up against anti-Semitism, everywhere.”
The New York Police Department said it had deployed officers to synagogues throughout the city as a precaution.
Image copyright NYPDCT Image caption Armed police are guarding synagogues in New York
The BBC’s Dan Johnson in Washington says the shootings come at a tense time in the US, after a week in which mail bombs were sent to critics of Mr Trump, ahead of crucial mid-term elections next month.
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