A group is called residually finite if for every non-identity element there is a normal subgroup of finite index which does not contain that element. One can easily show that residually finite groups embed in their profinite completion as dense subgroups. In this talk we will discuss the following problem: Does there exists an infinite residually finite group in which every centralizer is finite? We will present recent results of this ongoing study joint with Maria Tota and Antonio Tortora.