In this thesis we study the maximum number of limit cycles that can bifurcate from a focus singular point of an analytic, autonomous differential system in the real plane under an analytic perturbation. We consider eing a focus singular point of the following three types: non-degenerate, degenerate without characteristic directions and nilpotent. In a neighborhood of the differential system can always be brought, by means of a change to (generalized) polar coordinates ( r, ? ), to an equation over a cylinder in which the singular point corresponds to a limit cycle . This equation over the cylinder always has an inverse integrating factor which is smooth and non–flat in r in a neighborhood of . We define the notion of vanishing multiplicity of the inverse integrating factor over . This vanishing multi- plicity determines the maximum number of limit cycles that bifurcate from the singular point in the non-degenerate case and a lower bound for the cyclicity otherwise. Moreover, we prove the existence of an inverse integrating factor in a neighborhood of many types of singular points, namely for the three types of focus considered in the previous paragraph and for any isolated singular point with at least one non-zero eigenvalue. The relation between limit cycles of planar differential systems and the inverse integrating factor was first shown in an article of Giacomini, Llibre and Viano appeared in 1996. From that moment on, many research articles are devoted to the study of the properties of the inverse integrating factor and its relation with limit cycles and their bifurcations. This thesis is a summary of all the results about this topic. We include a list of references together with the corresponding related results aiming at being as much exhaustive as possible. The thesis is, nonetheless, self-contained in such a way that all the main results on the inverse integrating factor are stated and a complete overview of the subject is given.