Abstract: A bi-Hamiltonian structure is a pair of Poisson structures which are compatible, meaning that their sum is again a Poisson structure. In contrast to individual Poisson structures, which can be always brought to a constant form near a generic point, bi-Hamiltonian structures have non-trivial local geometry. I will try to give a survey talk on this matter. In particular, I will discuss the relation of bi-Hamiltonian structures to Veronese webs and classical webs. I will also mention some results of mine on flat bi-Hamiltonian structures.