We present a general dynamical theory of a membrane coupled to an actin cortex containing polymerizing filaments with active stresses and currents, and demonstrate that active membrane dynamics [Phys. Rev. Lett \textbf{84}, 3494 (2000)] and spontaneous shape oscillations emerge from this description. We also consider membrane instabilities and patterns induced by the presence of filaments with polar orientational correlations in the tangent plane of the membrane. The dynamical features we predict should be seen in a variety of cellular contexts involving the dynamics of the membrane-cytoskeleton composite and cytoskeletal extracts coupled to synthetic vesicles.