Abstract: Based on Simo and Fox's well-known work, this article gives a comprehensiveoverview of the single director shell model and its modern representation.It describes the mathematical foundation and the mechanicalanalysis of a fully nonlinear, geometrically exact shell theory. Thisformulation of the shell equations has the advantage that, although theexact nonlinear geometric structures of the idealized two-dimensional bodyare maintained, the complex differential geometry concepts such as covariantderivatives, Riemannian connections or Christoffel symbols are avoided.Furthermore, a weak or variational statement of the local balance equationsis developed, as is ideally suited for numerical solution techniques.