LATEST from our reporting:
In a video posted on the Internet, a woman identified as the director of the college, Olga Grebennikova, says there were "many corpses, many corpses of children" and calls it "a real terrorist act."
The woman, speaking on a mobile phone, says that shortly after she left the college for a meeting elsewhere, assailants attacked the school, throwing explosive devices and firing guns.
"They ran up to the second floor with automatic rifles -- I don't know with what -- and opened doors...and killed everyone they could find," she said. She said students and faculty were among the dead.
The authenticity of the video could not immediately be independently confirmed.
President Vladimir Putin's spokesman, Dmitry Peskov, said Russian authorities were considering terrorism as a possible motive.
Putin voiced condolences to victims and their loved ones and ordered the authorities to evacuate badly injured victims by air to "leading" hospitals in Moscow and other cities, Peskov said.
Video from a local media outlet showed ambulance crews helping victims after a blast which Russia said was caused by an explosive device. Initial reports said 10 people were feared dead and dozens injured by the blast in Kerch, Crimea, which Russia seized from Ukraine in 2014.
