  We construct a new cohomology functor from the a certain category of {\it
quantum operator algebras} to the category of {\it Batalin-Vilkovisky
algebras}. This {\it Moonshine cohomology} has, as a group of natural
automorphisms, the Fischer-Griess Monster finite group. We prove a general
vanishing theorem for this cohomology. For a certain commutative QOA attached
to a rank two hyperbolic lattice, we show that the degree one cohomology is
isomorphic to the so-called Lie algebra of physical states. In the case of a
rank two unimodular lattice, the degree one cohomology gives a new construction
of Borcherd's Monster Lie algebra. As applications, we compute the graded
dimensions and signatures of this cohomology as a hermitean Lie algebra graded
by a hyperbolic lattice. In the first half of this paper, we give as
preparations an exposition of the theory of quantum operator algebras. Some of
the results here were announced in lectures given by the first author at the
Research Institute for Mathematical Sciences in Kyoto in September 94.
