Groups with small cocentralizers

Let $G$ be a group. If $S\subseteq G$ is a $G$–invariant subset of $G$, the factor-group $G/C_G(S)$ is called the cocentralizer of $S$ in $G$. In this survey-paper we review some results dealing with the influence of several cocentralizers on the structure of the group, a direction of research to which Leonid A. Kurdachenko was an active contributor, as well as many mathematicians all around the world.