The alleged Florida State University shooter stalked his prey around the campus, firing at people as they ran for their lives, and it was caught on terrifying video.

Check it out ... suspected gunman Phoenix Ikner is allegedly seen in the footage walking down a path before he abruptly stops, turns and opens fire at screaming students, who scatter in all directions. A person appears to be lying on the ground nearby.

Another clip shows a bunch of cops carrying what appears to be a wounded shooting victim across the campus during the rampage.

You can hear someone recording the video express horror and shock, saying, "Oh my God." A frightened person nearby wonders out loud if the alleged victim was shot.

Other videos capture students hiding under desks in a classroom, and running from the FSU Tallahassee grounds, with one young woman asking if everyone is OK.

Authorities say Ikner is the stepson of Leon County Sheriff's deputy Jessica Ikner -- and had access to his mom's handgun, which he allegedly used during the violent spree ... along with a shotgun.

Cops say Ikner fatally shot 2 people, who were not students, and injured 5 more, before being shot by the police and taken into custody. He was also hospitalized.

Ikner is believed to be a student at FSU ... investigators have not nailed down a motive for the shooting. Ikner has not been formally charged.