This chapter treats a range of problems that can be solved explicitly within the framework of nonlinear elasticity. We begin with deformations used in material testing, including uni-axial tension, simple shear, and torsion, and show how the experimental data can help identify a suitable model for a given soft solid such as rubber or brain matter. We also pay special attention to energy methods, which prove to be powerful tools for resolving complex deformation problems such as the Poynting effect in torsion and the inflation of spherical membranes. We then discuss the homogeneous deformations of electroactive membranes under coupled mechanical and electric loads and their stability. We conclude with an exploration of the Biot instability, which models the formation of wrinkles on elastic surfaces under critical compression.