Given a variety V defined over the field of rational numbers, a height function is a tool designed to study the "size" or the "complexity" of the rational points on V. When the set of rational points of V is non-empty, it is natural to ask what can be said about the minimal height of a rational point on V. In this talk I will make this question more precise and I will present results dealing with the case of families of Fano hypersurfaces.