  Monstrous moonshine relates distinguished modular functions to the
representation theory of the monster. The celebrated observations that
196884=1+196883 and 21493760=1+196883+21296876, etc., illustrate the case of
the modular function j-744, whose coefficients turn out to be sums of the
dimensions of the 194 irreducible representations of the monster. Such formulas
are dictated by the structure of the graded monstrous moonshine modules. Recent
works in moonshine suggest deep relations between number theory and physics.
Number theoretic Kloosterman sums have reappeared in quantum gravity, and mock
modular forms have emerged as candidates for the computation of black hole
degeneracies. This paper is a survey of past and present research on moonshine.
We also compute the quantum dimensions of the monster orbifold, and obtain
exact formulas for the multiplicities of the irreducible components of the
moonshine modules. These formulas imply that such multiplicities are
asymptotically proportional to dimensions.
