One of the open problems in the theory of algebraic cycles is that of the existence of Künneth projectors for any smooth, projective algebraic variety X: in other words, of algebraic cycles on X \times X, acting on (singular, étale...) cohomology H*(X) as the projections on each H^i(X). This problem admits a version in families: given a projective, non necessarily smooth morphism f:X -> S between varieties, one asks whether there exist algebraic cycles decomposing the complex Rf_* 1_X into simple factors, called intersection complexes. The modern theory of motives allows one to study these problems by constructing, under suitable hypotheses, certain intersection motives, analogues of their sheaf-theoretic homonyms. The aim of the talk is to present an overview of the applications of these objects in several geometric contexts: quadric bundles, hyper-Kähler varieties, Shimura varieties.