We first present a short review of general supersymmetric compactifications in string and M-theory using the language of G-structures and intrinsic torsion. We then summarize recent work on the generic conditions for supersymmetric AdS5 backgrounds in M-theory and the construction of classes of new solutions. Turning to AdS5 compactifications in type IIB, we summarize the construction of an infinite class of new Sasaki–Einstein manifolds in dimension 2k + 3 given a positive curvature Kähler–Einstein base manifold in dimension 2k. For k = 1 these describe new supergravity duals for N = 1 superconformal field theories with both rational and irrational R-charges and central charge. We also present a generalization of this construction, that has not appeared elsewhere in the literature, to the case where the base is a product of Kähler–Einstein manifolds.