The concept was first introduced by David Chaum and Eugene van Heyst in 1991.
Another application is for keycard access to restricted areas where it is inappropriate to track individual employee's movements, but necessary to secure areas to only employees in the group.
Boneh, Boyen and Shacham published in 2004 (BBS04, Crypto04) a novel group signature scheme based on bilinear maps.
The security of the scheme is proven in the random oracle model and relies on the Strong Diffie Hellman assumption (SDH) and a new assumption in bilinear groups called the Decision linear assumption (DLin).
A more formal definition that is geared towards provable security was given by Bellare, Micciancio and Warinschi.