Given a (smooth) projective (complex) surface S and a complete linear (or algebraic) system of curves on S, one defines the Severi varieties to be the (possibly empty) subvarieties parametrizing nodal curves in the linear system, for any prescribed number of nodes. These were originally studied by Severi in the case of the projective plane. Afterwards, Severi varieties on other surfaces have been studied, mostly rational surfaces, K3 surfaces and abelian surfaces, often in connection with enumerative formulas computing their degrees. Interesting questions are nonemptiness, dimension, smoothness and irreducibility of Severi varieties. In this talk I will first give a general overview and then present recent results about Severi varieties on Enriques surfaces, obtained with Ciliberto, Dedieu and Galati, and the connection to a conjecture of Pandharipande and Schmitt.