Let X be a smooth projective surface and choose a curve C on X. Let VC be the set of all irreducible divisors on X linearly equivalent to C whose normalization is a rational curve. The Severi problem for rational curves on X with divisor class [C] consists of studying the irreducibility of the spaces VC as C varies among all curves on X. In this talk, we will prove that all the spaces VC are irreducible in the case where X is a del Pezzo surface, with a unique exception.