I can never remember the publication dates of the various versions of RuneQuest, so here's a list:
RuneQuest (1978), Chaosium. AKA RuneQuest I, RQI, RQ1. Was set in the world of Glorantha.
RuneQuest, second edition (1980), Chaosium. AKA RuneQuest II, RQII, RQ2. Was set in the world of Glorantha.
RuneQuest, third edition (1984), Avalon Hill. AKA RuneQuest III, RQ3. Used Fantasy Europe as the example setting, but also had a lot of Glorantha publications.
Originally published as a box set, it was republished as a paperback book in the US in the early 1990s during the brief but productive RuneQuest Renaissance ushered in by the new RuneQuest editor at Avalon Hill, Ken Rolston.
RuneQuest: Adventures in Glorantha, (never published commercially), Avalon Hill. AKA RQIV and RQ:AiG on the Internet.
RuneQuest (2006), Mongoose Publishing. AKA MRQ on the Internet.
This was published as an setting-independent fantasy system (under the OGL), but Mongoose also had a license to publish material about the “Second Age” of the world of Glorantha and published a lot of Glorantha specific books.
RuneQuest, second edition (2010), Mongoose Publishing. AKA MRQ2 on the Internet. (Mongoose called this “RuneQuest II”, and books for this edition usually have a RQII logo on their cover.)
RuneQuest, sixth edition (announced 16 July 2011), The Design Mechanism.
I'll not go into the related histories of the related Basic Roleplaying games from Chaosium, nor the completely different system HeroQuest which also used the setting of Glorantha.
Mongoose released a new RPG, Legend, using the mechanics from their RuneQuest, second edition, under the OGL.