And, Yes, They Do Really Fly -

Tree Of Life Web:

Members of the Ommastrephidae are small (about 10 cm ML) to large (about 100 cm ML), muscular squids that are often the dominant large squids in oceanic and, occasionally, neritic waters. A number of species are fished commercially.

Ommastrephid squids are among the strongest swimmers in the Cephalopoda. Some are commonly known as "flying squid" due to their ability to glide over the ocean surface as seen in the photographs.