Hmm.....
Fighters: Street Fighter, (SF2 will always be my favourite game of all time) Dead or alive, Blazblue, Soul Calibur, World Heroes, Vs Series etc. (Though I'll admit I really suck at these games now.)
Action JRPG's Tales series love them, I actually like FF13 except still have played Lightning Returns will get it for PC I guess, can't get tin Western RPG's, (largely because I tried a few and every single one managed to crash on me, early on, I must be cursed.
Arcade Racers: Sega Rally, (that first course on the Saturn Version has to be my favourite course ever), Need for Speed, and Burnout.
Shooters: Halo, Destiny, and Borderlands.
Open world: GTA, Red Dead Redemption, etc.
like the odd Simulations type games, Sim City, Football Manager etc.