  We review the remarkable progress that has been made the last 15 years
towards the classification of supersymmetric solutions with emphasis on the
description of the bilinears and spinorial geometry methods. We describe in
detail the geometry of backgrounds of key supergravity theories, which have
applications in the context of black holes, string theory, M-theory and the
AdS/CFT correspondence unveiling a plethora of existence and uniqueness
theorems. Some other aspects of supersymmetric solutions like the Killing
superalgebras and the homogeneity theorem are also presented, and the
non-existence theorem for certain smooth supergravity flux compactifications is
outlined. Amongst the applications described is the proof of the emergence of
conformal symmetry near black hole horizons and the classification of warped
AdS backgrounds that preserve more than 16 supersymmetries.
