The accident is thought to have happened when one tram ploughed into the rear of another after it had stopped at a traffic light.
One of the trams was derailed in the collision, which happened on a busy route in South Yarra shortly after 10.30am.
The driver of one of the trams is said to be in a serious condition, after he was trapped inside the vehicle.
Between 10 and 12 others have been taken to hospital with minor to moderate injuries. All are said to be in a stable condition.
Fire crews and ambulances were still at the scene, on St Kilda Road, at the junction of Leopold Street, several hours after the crash.
"There's quite a large contingent of paramedics on the scene as well as a large number of police," said Yarra Trams spokesman Colin Tyrus.
"Passengers are not trapped, they are being attended to by ambulance officers, [the passengers are] in neck braces, obviously there's some type of impact injuries or whiplash injuries," he said.
"The only person trapped was the driver of the rear tram."
The trams involved were the number 6 and the 72, both of which had been heading to the city and the university.
Mr Tyrus said the the Department of Infrastructure and Yarra Trams would investigate the cause of the crash, but at this stage he did not think high speeds were involved.
"Obviously the speed will be checked when we collect the data records from the trams," he said.
"One [tram] had just departed from a stop light at the intersection, so it was starting to accelerate when the other tram collided with it."
Traffic in the area was diverted, and a long backlog of trams built up.