The first one was called
San Francisco Rush: Extreme Racing.
I played this on N64. Was quite fun! Big jumps! It was a good port from the arcade machine.
There was a
Rush 2: Extreme Racing USA later. Haven't played it, but always wanted to.
There was also a third one:
San Francisco Rush 2049. But I don't know if it ever came to Europe. But it was also ported to GBA and Dreamcast.
After that I don't know how the series moved on. I think there was
L.A. Rush on PS2, but I dunno.