  We study the RG flow of two dimensional (fluid) membranes embedded in
Euclidean D-dimensional space using functional RG methods based on the
effective average action. By considering a truncation ansatz for the effective
average action with both extrinsic and intrinsic curvature terms we derive a
system of beta functions for the running surface tension, bending rigidity and
Gaussian rigidity. We look for non-trivial fixed points but we find no evidence
for a crumpling transition at $T\neq0$. Finally, we propose to identify the
$D\rightarrow 0$ limit of the theory with two dimensional quantum gravity. In
this limit we derive new beta functions for both cosmological and Newton's
constants.
