The existence theory in nonlinear elasticity, beginning with the pioneering work of John Ball in the seventies, constitutes one of the major achievements in the nonconvex calculus of variations. A central role is played by the Jacobian determinant, a highly nonlinear function of the gradient that has, nonetheless, a distinct divergence structure that encodes the remarkable fusion between topology, differential geometry, and mechanics that governs the response of an elastic body and that makes it possible to succeed in the analysis, sharing with homogenization theory the use of compensated compactness. This fragile balance is severely confronted if deformations large enough to induce or propagate fracture are present. In this case, singular structures are formed, which paradoxically produce exceedingly large values of the energy density being minimized. A suitable geometric characterization of these structures, based on the theory of functions of bounded variation and on Federer-Fleming’s closure theorem for integral currents, is in order. The methods developed have implications for the analysis of magnetoelasticity and nematic elastomers, as well as a striking connection with harmonic map theory because of the dipoles emerging in the borderline neo-Hookean open problem.