  We explore a new class of supersymmetric models for lepton masses and mixing
angles where the role of flavour symmetry is played by modular invariance. The
building blocks are modular forms of level N and matter supermultiplets, both
transforming in representations of a finite discrete group Gamma_N. In the
simplest version of these models, Yukawa couplings are just modular forms and
the only source of flavour symmetry breaking is the vacuum expectation value of
a single complex field, the modulus. In the special case where modular forms
are constant functions the whole construction collapses to a supersymmetric
flavour model invariant under Gamma_N, the case treated so far in the
literature. The framework has a number of appealing features. Flavon fields
other than the modulus might not be needed. Neutrino masses and mixing angles
are simultaneously constrained by the modular symmetry. As long as
supersymmetry is exact, modular invariance determines all higher-dimensional
operators in the superpotential. We discuss the general framework and we
provide complete examples of the new construction. The most economical model
predicts neutrino mass ratios, lepton mixing angles, Dirac and Majorana phases
uniquely in terms of the modulus vacuum expectation value, with all the
parameters except one within the experimentally allowed range. As a byproduct
of the general formalism we extend the notion of non-linearly realised
symmetries to the discrete case.
