When intense short-pulse laser beams (I > 1022 W/m2, τ < 20 ps) interact with high density plasmas, strong shock waves are launched. These shock waves may be generated by a range of processes, and the relative significance of the various mechanisms driving the formation of these shock waves is not well understood. It is challenging to obtain experimental data on shock waves near the focus of such intense laser–plasma interactions. The hydrodynamics of such interactions is, however, of great importance to fast ignition based inertial confinement fusion schemes as it places limits upon the time available for depositing energy in the compressed fuel, and thereby directly affects the laser requirements. In this manuscript we present the results of magnetohydrodynamic simulations showing the formation of shock waves under such conditions, driven by the j × B force and the thermal pressure gradient (where j is the current density and B the magnetic field strength). The time it takes for shock waves to form is evaluated over a wide range of material and current densities. It is shown that the formation of intense relativistic electron current driven shock waves and other related hydrodynamic phenomena may be expected over time scales of relevance to intense laser–plasma experiments and the fast ignition approach to inertial confinement fusion. A newly emerging technique for studying such interactions is also discussed. This approach is based upon Doppler spectroscopy and offers promise for investigating early time shock wave hydrodynamics launched by intense laser pulses.