Severi varieties parametrize integral curves of fixed geometric genus in a given linear system on a surface. In this talk, I will present some results on the irreducibility of these varieties in the case of toric surfaces, and their application to the irreducibility of other moduli spaces of curves. This is done using tropical methods and I will indicate some of these aspects. The new results are from ongoing joint work with Xiang He and Ilya Tyomkin.