We discuss recent experimental work demonstrating spatial dispersive shock waves (DSWs). These structures occur whenever nonlinearity enhances diffraction so that wave spreading becomes intensity-dependent. The mechanism of this spreading follows naturally from a hydrodynamic description of light flow, in which wave steepening from phase gradients allows faster parts of a beam to overtake slower parts. Scaling relationships are developed for this spreading and experimentally observed, in both local and nonlocal media. DSWs are a fundamental means of energy transport and arise whenever a potential disturbs the ambient flow. For a single barrier, both transmitted and reflected waves give rise to shocks. For a periodic array of barriers, the underlying Bloch-mode structure gives rise to lattice shock waves. These beams are the exact opposite of lattice solitons: instead of a propagation constant residing in a band gap, isolated from other modes, the propagation constant is pushed further into a band, facilitating transport across the array. In terms of applications, spatial DSWs hold potential as optical limiters and as nonlinear point-spread functions in imaging. For known media, matching shock measurements with model properties allows calibration of the propagation parameters. For unknown media, observations of the nonlinear transport facilitate new models of the optical response. Examples are given for thermal media and for photorefractive crystals.