A general nonlinear theory for thin shells of arbitrary midsurface geometry is formulated in terms of a finite rotation vector and a stress-function vector. Compatibility equations, equilibrium equations, and boundary conditions are derived which are valid for shells undergoing arbitrarily large rotations and strains. For problems admitting a potential energy functional, a variational principle is formulated. The simplifications implied by small extensional strains are discussed. The theory contains, as special cases, Reissner’s equations for the axisymmetric deformation of shells of revolution, and the Sanders-Koiter linear shell theory.