<p>Disordered media include metallic glasses, colloidal suspensions, granular matter and biological tissues, among others. Their physics offers difficult challenges because it often occurs far from equilibrium, in materials that lack symmetries and that evolve through complex energy landscapes. We review theoretical efforts from recent years to provide microscopic insights into the mechanical properties of amorphous media using approaches from statistical mechanics as unifying frameworks. Our focus is on how amorphous solids become unstable and yield under applied deformations. We cover both the initial regime, corresponding to small deformations of the solid, and the transition between elastic response and plastic flow when the solid yields. We discuss the specific features arising for systems evolving near a jamming transition and extend our discussion to recent studies of the rheology of dense biological and active materials. We emphasize the importance of a unified approach to studying the response to deformation and the yielding instability of a broad range of disordered media.</p>