The concept of $p$-local finite group arise in the work of Broto-Levi-Oliver as a generalization of the classical concept of finite group. Therefore, the classification of $p$-local finite groups has interest, not only by itself but, as an opportunity to enlighten one of the highest mathematical achievements in the last decades: The Classification of Finite Simple Groups.

   In this work we classify the $p$-local finite groups over rank two $p$-groups. This study is divided in two parts: first we consider the $p$-local finite groups over $p^{1+2}_+$, where we obtain three new exotic $7$-local finite groups, and secondly we consider the other cases, where we check that the fusion is controlled by the normalizer of the Sylow $p$-subgroup.

   (join work with Antonio DÌaz and Antonio Viruel)