A modular graph function maps a Feynman graph for a conformal scalar field in the plane to a modular function. Modular graph functions generalize non-holomorphic Eisenstein series for genus one, and Kawazumi-Zhang and Faltings invariants for higher genus. They arise naturally in string theory, where their integral over the moduli space of compact Riemann surfaces determines the strength of the effective interactions in the low energy expansion of string theory. They satisfy remarkable algebraic and differential identities.