  We present and analyze solutions of D=11 supergravity describing the
``near-horizon'' (i.e., asymptotically AdS_4 x S^7) geometry of M2-branes
wrapped on surfaces of arbitrary genus. We study the forces experienced by test
M2-branes in such backgrounds, and find evidence that extremal branes on
surfaces of genera higher than the torus are unstable. Using the holographic
connection between AdS spaces and superconformal field theories in the large N
limit, we discuss the phases of the associated 2+1 dimensional theories.
Finally, we also study the extension of these solutions to other branes, in
particular to D2-branes.
