The Picard–Fuchs differential equation of a family of varieties is useful for describing the variation of Hodge structure of the family. I will give the definition of the Picard–Fuchs equation in general, and describe its computation. I will give examples for the case of families of elliptic curves and families of K3 surfaces.

I will concentrate on the case of modular families of of elliptic curves, which leads to differential equations for modular forms, and if there is time I will also describe a relationship between a solution of the Picard-Fuchs equation and equation the L-series of the ambient space in certain cases.