An analytical theory is developed that describes the process of nonlinear saturation of the magnetorotational instability. The theory employs the shearing sheet approximation with arbitrary (but linear) shear flow, and is asymptotically exact in a well-defined parameter regime. In this regime the instability saturates by extracting energy from the shear, and modifying the shear responsible for it. The saturation process requires both viscous and ohmic dissipation. The theory also describes the approach from small amplitude perturbations to the final strongly nonlinear saturated state.