Landau damping is a fundamental phenomenon in plasma physics, which also plays an important role in astrophysics, and sometimes under different names, in fluid dynamics and other fields. Its theoretical discussion in the framework of the Vlasov equation often assumes that the reference stationary state is homoge-neous in space. However, Landau damping around an inhomogeneous reference stationary state, a natural setting in astrophysics for instance, induces new mathematical difficulties and physical phenomena. The goal of this article is provide an introduction to these problems and the questions they raise
