In this lecture we will review some classic and more recent results on the class of log-concave functions, focusing on the analogies with the theory of convex bodies. We will be particularly interested in functional inequalities. The main example will be the Prékopa-Leindler inequality, that we will present along with its "infinitesimal" version; but we will also see other examples like the functional versions of the Blashke-Santaló inequality and of Rogers-Shephard inequality. Another topic that we will touch is the possibility to extend to these functions the notions of intrinsic and mixed volumes, as well as the study of valuations.