  We prove the renormalizability of the generalized Edwards model for
self-avoiding polymerized membranes. This is done by use of a short distance
multilocal operator product expansion, which extends the methods of local field
theories to a large class of models with non-local singular interactions. This
ensures the existence of scaling laws for crumpled self-avoiding membranes, and
validates the direct renormalization method used for polymers and membranes.
This also provides a framework for explicit perturbative calculations. We
discuss hyperscaling relations for the configuration exponent and contact
exponents. We finally consider membranes with long range interactions and at
the Theta-point.
