Fallout series, Jade empire, Dreamfall: The longest journey, Fahrenheit: The indigo prophecy (get directors ed. so you can see boobies), The Curse of Monkey island, Planescape Torment, Baldurs gate series, and last but not least, fable.
I could go on for hours, there are a ton of good storylines out there, you just have to know where to look.
I think this was released as Fahrenheit in the EU/UK in case you have trouble finding it, and was quite a well made game. The whole playing as the person running and person chasing thing is a great concept and was recently reminded of how decent the gameplay can be with this idea when I started playing through Call of Juarez again, not a fantastic story as such but told from 2 viewpoints which cross paths and coincide a lot which make the storytelling suprisingly enjoyable.
Mafia has an excelent storyline (mostly from what I hear, I never got very far on it)
Fallout 1 and 2 are good, it's not what I'd call a story as such but more of an adventure/experience, you get to the end and you've sort of created your own story based around the ways you tackled your missions and the things you did.
my fav's for storyline morrowing goty (game of the year), final fantasy 7 if you havent played it, metal gear tes, fear, grandia, zelda ocarina of time, zelda twilight princess, or most of the castlevanias