  We construct a gauge fixed action for topological membranes on $G_2$-manifold
such that its bosonic part is the standard membrane theory in a particular
gauge.
  We prove that quantum mechanically the path-integral in this gauge localizes
on associative submanifolds. Moreover on $M\times S^1$ the theory naturally
reduces to the standard A-model on Calabi-Yau manifold and to a membrane theory
localized on special Lagrangian submanifolds. We discuss some properties of
topological membrane theory on $G_2$-manifolds. We also generalize our
construction to topological $p$--branes on special manifolds by exploring a
relation between vector cross product structures and TFTs.
