In this thesis the numerical simulation of the structural and stability behavior of shells is adressed. A theoretical description and the numerical implementation of a geometric and physically nonlinear finite shell element is given. Within the framework of the finite element method, numerical procedures for the computation of the buckling and post-buckling behavior of stiffened shell structures are presented and evaluated. After summarizing the basic continuum mechanics equations, a geometric exact shell theory is obtained via the standard spatial reduction process. The variational basis for the element formulation is the Hu-Washizu functional with independent displacement, strain and stress field. The reference surface is discretized by a four node quadrilateral element. To avoid transverse shear locking, the ANS concept is applied. The interpolation of the independent strain field is split into two steps. The first step is identical to the interpolation of the stress field. In analogy to the EAS concept, in the second step interpolation orthogonal to the stress field is conducted. Thickness strains are also considered in this step. Thus, arbitrary three-dimensional material models can be utilized without modification. This interpolation scheme renders a stiffness matrix of correct rank. Complete load-displacement curves are employed to assess the stability behavior of the considered shell. Hence, so-called direct methods are used to determine (primarily) bifurcation points. These methods are based on equilibrium equations that are extended by additional conditions, describing the singularity of the stiffness matrix. To avoid the singularity of the Jacobian matrix close to the bifurcation point, a modification of the extended equilibrium equations is suggested. Based on the proposed method, the nonlinear structural response of several shell problems is predicted numerically.