Hyper-Kähler manifolds are symplectic holomorphic compact K¨ahler manifolds, a particular class of complex manifolds with trivial canonical bundle. They form a very restricted class, both from the Riemannian and complex geometry viewpoints. They also satisfy strong topological restrictions that I will describe, and that follow from the study of their deformations. I will also explain further restrictions in dimension 4, and a classification result obtained in collaboration with Debarre, Huybrechts and Macrì.