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We study in detail the structure of the Lorentz covariant, spacetime supersymmetric ll-dimensional supermembrane theory. We show that for a flat spacetime background, the spacetime supersymmetry becomes an N =8 world volume (rigid) supersymmetry in a “physical ” gauge; we also present the field equations and transformation rules in a “lightcone” gauge. We semiclassically quantize the closed torodial supermembrane on a spacetime (Minkowki), x (flu / ‘I-torus), and review some mathematical results that are relevant for path integral quantization. ‘cl 1988 Academic Press, Inc. 1