  We investigate nonperturbative effects in M-theory compactifications arising
from wrapped membranes. In particular, we show that in $d=4, \mathcal{N}=1$
compactifications along manifolds of $G_2$ holonomy, membranes wrapped on rigid
supersymmetric 3-cycles induce nonzero corrections to the superpotential. Thus,
membrane instantons destabilize many M-theory compactifications. Our
computation shows that the low energy description of membrane physics is
usefully described in terms of three-dimensional topological field theories,
and the superpotential is expressed in terms of topological invariants of the
3-cycle. We discuss briefly some applications of these results. For example,
using mirror symmetry we derive a counting formula for supersymmetric
three-cycles in certain Calabi-Yau manifolds.
