The most accessible problems for the mechanics of deformable solid bodies are those for thin bodies, namely, rods and shells, because their equations respectively have but one and two independent spatial variables. There is a voluminous literature devoted to the derivations of various models for such bodies undergoing small deformations. On the other hand, geometrically exact theories are derived directly from fundamental principles. They readily accommodate general nonlinear material response. This lecture describes solutions of a variety of steady-state and dynamical geometrically exact problems, emphasizing the appearance of thresholds in constitutive response that separate qualitatively different behaviors.