We present the main properties of Sobolev spaces: these are functional spaces made of functions which are differentiable in weak sense. We pay due attention to the space of Sobolev functions “vanishing at the boundary”, in a weak sense. The chapter contains in particular: the main functional inequalities for Sobolev functions (including Sobolev, Ladyzhenskaya and Morrey inequalities, as well as various versions of Poincaré’s inequality and Hardy inequality), embedding theorems, operations on Sobolev functions and a brief discussion on extension operators.