This paper opens with a general discussion of terms in an energy functional which might be the basis from which equations governing stress, stability, and vibration analyses are derived. The energy expression includes strain energy of the shell and discrete stiffeners, kinetic energy of the shell and stiffeners, constraint conditions with Lagrange multipliers, and other terms arising from the change in direction of applied loads during deformation. Brief discussions are included of the coupling effect between bending and extensional energy needed for the analysis of layered composite shells or elastic-plastic shells, nonlinear terms, and the form that the energy expression takes upon discretization of the structure. A section follows in which the energy formulation for stress, stability, and vibration analyses of an elastic curved beam is given, including thermal effects, moderately large rotations, boundary conditions, and distributed and concentrated loads. The matrix notation and type of discretization are introduced here which will later be used for the analysis of shells of revolution. Terms in the local element stiffness, mass, and load-geometric matrices are derived in terms of nodal point displacements, and it is shown how these local matrices are assembled into global matrices. The purpose of the section is to demonstrate the procedure for derivation of the analogous equations and quantities for shells of revolution or more complex structures. The next section is on elastic shells of revolution. It opens with a summary of what computer programs exist for stress, buckling, and stability analyses of such structures. The assumptions on which these programs are based are listed and the various components of the energy functional, such as strain energy of the shell and discrete rings, are identified and derived in terms of nodal point displacements. Included are a derivation of the constitutive law for anisotropic shell walls and a formulation of nonlinear constraint conditions, which are required for the treatment of segmented or branched shells with meridional discontinuities between segments or branches. Derivations of terms in the global stiffness and load-geometric matrices and the force vector are given, with tables tracing the origin of each term. The computational strategy for calculation of critical bifurcation buckling loads in the presence of prebuckling nonlinearities is given, with an example of buckling under axial compression of a very thin cylinder. This is a simple problem to formulate but a difficult one to solve numerically, owing to the existence of closely spaced eigenvalues corresponding to nonsymmetric buckling at loads close to the load corresponding to nonlinear axisymmetric collapse. A description of various pitfalls encountered in the search for the lowest bifurcation buckling load is given, including estimates of the critical number of circumferential waves in the buckling mode. Computerized formulations and run times are compared for various discretization methods, including finite difference energy models and standard finite element models, with an example showing comparisons of rate of convergence with increasing nodal point density and computer times required to form stiffness matrices. Hybrid bodies of revolution are discussed next. By “hybrid” is meant a body of revolution with both one-dimensionally and two-dimensionally discretized regions. The formulation is particularly useful for the stress, buckling, and vibration analyses of branched shells or ring-stiffened shells in which one is particularly interested in local effects within a distance equal to a shell wall thickness of a branch or ring. An appropriate strategy for the solution of nonlinear problems with simultaneous geometric nonlinearity and path-dependent material properties is described, including the development of the incremental constitutive law for the tangent stiffness method of treatment of elastic-plastic structures. The two-dimensionally discretized regions are modeled with use of 8-node isoparametric quadrilaterals of revolution. Details are presented on the formulation of constraint conditions for compatibility at junctions between rotationally symmetric shell segments (one-dimensionally discretized regions) and solid segments (two-dimensionally discretized regions). The paper closes with a summary of linear equations for general shells. Surface coordinates, the first and second fundamental forms, and the definition of a shell are introduced, and the assumptions corresponding to Love's first approximation are identified. The differences in commonly used or referenced formulations are listed, including differences with regard to kinematic relations, expressions for total strain anywhere in the thickness of the shell wall, and expressions for stress and moment resultants. Comments are offered on which theory is the most suitable for engineering estimates.