Lectures on Mean Curvature Flow - Carlo Mantegazza In this series of lectures I will introduce the mean curvature flow of a compact hypersurface in the Euclidean space with particular attention to the cases of curves and surfaces. The basic properties and the main analytic and geometric techniques used in the analysis of this flow will be discussed, for instance maximum and comparison principles. Moreover, I will present the fundamental Huisken.s monotonicity formula and the Harnack inequality by Hamilton, which are the key tools in the study of the singularity formation. Up to now, the classification of possible asymptotic shape of a singularity is almost complete for some classes of evolving hypersurfaces. For others it seems difficult and quite far off. Time allowing, in the last ectures I will try to outline an up.to.date scenario of the "state of the art".