  It is suggested that topological membranes play a fundamental role in the
recently proposed topological M-theory. We formulate a topological theory of
membranes wrapping associative three-cycles in a seven-dimensional target space
with G_2 holonomy. The topological BRST rules and BRST invariant action are
constructed via the Mathai-Quillen formalism. In a certain gauge we show this
theory to be equivalent to a membrane theory with two BRST charges found by
Beasley and Witten. We argue that at the quantum level an additional
topological term should be included in the action, which measures the
contributions of membrane instantons. We construct a set of local and non-local
observables for the topological membrane theory. As the BRST cohomology of
local operators turns out to be isomorphic to the de Rham cohomology of the G_2
manifold, our observables agree with the spectrum of d=4, N=1 G_2
compactifications of M-theory.
