Virtua Fighter, and its sequel, were the definitive Sega Saturn fighting games for a very long time. Sega's AM2 division converted both from the arcade versions, and pulled off two very faithful conversions. The control system for both was identical: A for block, B for punch and C for kick. Unfortunately, due to the Saturn's lacklustre shoulder buttons, jumping was consigned to Up on the D-pad - rather clumsy, but workable nonetheless.
The original featured eight characters: Jacky, Jeffrey, Sarah, Kage, Pai, Wolf, Lau and Akira. Virtua Fighter 2 added two new characters to the mix, the ever-drunken Shun Li and the vaguely homoerotic French fighter Lion. AM2 had devised quite elaborate background stories for these characters, detailed in the manual: Jacky and Sarah were siblings; Pai was Lau's daughter; to name but a few. Every character had a huge array of special moves, kindly detailed in the manual, and this encouraged players to specialise in a few chosen characters.
Both games featured a straight-forward single player, lacking in any real story or progression, and neither was particularly challenging. However, a Ranking Mode was included, where players were scored for their variety and speed of attacks. This added some lasting appeal to solo player, but most entertainment was gleaned from multiplayer. Beating your friends in outrageous fashions never became boring, for me; perhaps due to the range of counter moves and easily constructable sequences of slick moves.
Virtua Fighter 3: Team Battle was released for DreamCast, but that didn't have quite the same appeal to me. I never bought it.
Virtua Fighter 4 is forthcoming on PlayStation 2, which looks set to recapture the graphical prowess and high speed combat that made the original enjoyable. Here's hoping.
Thanks to Saige for reminding me of Lau.