In this thesis, we are interested in the geometry of algebraic varieties that appear as minimal resolutions of quotients of the product of curves by the action of a finite group. We then study the positivity of their cotangent bundle because of its many geometric implications and the valuable and useful information that can be obtained in order to approach some difficult problems such as the famous conjectures of Lang, Lang-Vojta and Green-Griffiths-Lang which in particular give strong constraints on the distribution of the rational curves in varieties of general type. In the case of dimension two, we give a criterion for the positivity of the cotangent bundle and we study the algebraic hyperbolicity of product-quotient surfaces. These results apply to the case of product-quotient surfaces of general type with geometric genus, irregularity and second Segre number equal to zero, for which we prove effective versions of the conjectures mentioned above. More generally in higher dimension, we obtain a criterion for the positivity of the cotangent bundle in the case of smooth quotients and we study in detail the case of the symmetric products of curves.