Abstract : In this talk we will discuss the topological structure of the set of composition operators acting on the Hilbert space of Dirichlet series. In particular, we will focus on the two following questions: let $C_\varphi$ and $C_\psi$ be two composition operators. When do $C_\varphi$ and $C_\psi$ belong to the same component of the space of composition operators? When is $C_\varphi-C_\psi$ compact?