Abstract: We discuss the Grothendieck-Witt groups of group rings over the infinite cyclic and infinite dihedral groups, and derived variants thereof. We show that in all cases there is a Mayer-Vietoris type approximation and a more difficult Nil-type error term. We then prove a Nil-Nil theorem that relates the arising Nil-terms. Our results apply equally well to K- and L-theory and other functors satisfying suitable localization theorems. This is joint work with J. Davis (Indiana) and A. Huber (Augsburg).