  In 1939 Rademacher derived a conditionally convergent series expression for
the elliptic modular invariant, and used this expression- the first Rademacher
sum - to verify its modular invariance. By generalizing Rademacher's approach
we construct bases for the spaces of automorphic integrals of arbitrary even
integer weight, for all groups commensurable with the modular group. We use
these Rademacher sums to illuminate various aspects of the structure of the
spaces of automorphic integrals, including the actions of Hecke operators. We
obtain a new characterization of the discrete groups of monstrous moonshine in
terms of Rademacher sums, and we develop connections between Rademacher sums
and a family of monstrous Lie algebras recently introduced by Carnahan. Our
constructions suggest conjectures relating monstrous moonshine to a
distinguished family of chiral three dimensional quantum gravities, and
relating monstrous Lie algebras and their Verma modules to the second
quantization of this family of chiral three dimensional quantum gravities.
