This talk will discuss certain singularities that arise in the solution to boundary value problems involving the swelling of otherwise hyperelastic solids. In this setting, both non-uniform swelling and constrained swelling give rise to nonhomogeneous deformation in the absence of externally applied load. The standard singularities that are encountered in nonlinear elasticity may occur, such as cavitation. Additional singularities also arise, such as loss of smoothness associated with the concentration of deformation on singular surfaces.