With their respective hits, I'm kind of surprised these two haven't been mentioned so far:
U2 - The best concert I've ever seen was U2 in Houston in '92, with the Pixies as the opening act. (The Pixies were great also - Frank Black is a hoot.) I've seen U2 twice since then, and their live shows are always epic. I think I own every U2 album, plus a bunch of bootleg recordings. I believe they have sold over 150 million records, and have won 22 Grammy awards (more than any other band).
The Cure - I was also fortunate to see these guys in person toward the end of their mainstream popularity. Great show, and I've got most of their albums.