  Two groups are virtually isomorphic if they can be obtained one from the
other via a finite number of steps, where each step consists in taking a finite
extension or a finite index subgroup (or viceversa). Virtually isomorphic
groups are always quasi-isometric, and a group G is quasi-isometrically rigid
if every group quasi-isometric to G is virtually isomorphic to G.
  In this survey we describe quasi-isometric rigidity results for fundamental
groups of manifolds which can be decomposed into geometric pieces. After
stating by now classical results on lattices in semisimple Lie groups, we focus
on the class of fundamental groups of $3$-manifolds, and describe the behaviour
of quasi-isometries with respect to the Milnor-Kneser prime decomposition
(following Papasoglu and Whyte) and with respect to the JSJ decomposition
(following Kapovich and Leeb). We also discuss quasi-isometric rigidity results
for fundamental groups of higher dimensional graph manifolds, that were
recently defined by Lafont, Sisto and the author. Our main tools are the study
of geometric group actions and quasi-actions on Riemannian manifolds and on
trees of spaces, via the analysis of the induced actions on asymptotic cones.
