Most do. There are a few (they are far in between though) that does not involve much fighting, if you play your cards right. For example, I think that in Planescape: Torment you only had to resort to combat six times or so during the entire game...in such games it all depends on how you create your characters. Many games allow you to talk your way out of some situations rather than fighting. Depends on what type of character you would like to play.
And that is what is so good about RPGs. You can re-live the same storyline in different ways, try out new options and see the results. I think I sometimes live myself into the role of my PC (Player Character) too much, though.
What kind of fighting it is depends entirely on the game. Some action-RPGs rely on frenetic mouse-clicking, other are determined by set of rules and others are a mix of both. I personally prefer having a set of rules to go on, since I never liked having to win combat by killing my mouse.
Sometimes the other aspects of a game outweighs this, though. Take Morrowind, for example...an amazing game, very interesting storyline, lots to explore, nice atmosphere...but horrible combat system. I would even say that that is the worst part of the entire game system. Still you will fight through 75% of the game, and it is still enjoyable. It's a thin line the developers wander sometimes.
Maybe I rambled on a bit there (sorry about that!), but to summarize; yes, most RPGs involve lots of fighting but usually there are (player-determined) options to avoid fighting in certain parts of the games.