This thesis investigates the large deformations of hyperelastic membranes and shells. The static nonlinear behavior and possible instabilities of the membrane are both analyzed. A detailed experimental analysis was carried out involving cylindrical membranes and shells with different geometries and initial axial forces and the influence of the axial force and the internal pressure were investigated. An apparatus was developed to support vertically the extended structure while it is filled with air. The membranes and shells used in the experiments are composed of an isotropic, homogeneous and hyperelastic rubber, which is modeled as a Neo-Hookean incompressible material, described by a single elastic constant, or a Mooney-Rivlin or Ogden material, described by two elastic constants. Theses constants were obtained by comparing the experimental and numerical solutions for the structure under traction. The structure was discretized using appropriate membrane or shell finite elements and the resulting nonlinear equilibrium equations solved using the FE software ABAQUS. When the extended structure was filled with air, it was observed that the pressure increased initially as the volume increased until a certain critical value was reached, after which a bubble was suddenly formed along the structure and the internal pressure decreased markedly with increasing volume. The experimental results are, as shown in the thesis, in satisfactory agreement with the theory. A detailed parametric analysis was also carried out to study the influence of the initial traction and geometric parameters on the non-linear behavior and load carrying capacity of the structure. The influence of different types of local imperfections was also studied in detail.