Severi varieties are spaces which parametrize plane curves of fixed degree and geometric genus (with at worst nodal singularities), and the definition can be easily extended to projective surfaces other than the plane. In this talk, I will discuss the problem of proving that Severi varieties are irreducible (or of determining their irreducible components, since irreducibility fails for some surfaces). Although irreducibility is known to hold for Severi varieties of \( \mathbb{P}^2 \) and a few other rational surfaces, the problem remains open in most cases.