This thesis is concerned with the structure of shocks occurring in dense regions of molec?ular clouds. These shocks are associated with the outflows from young stars, Herbig-Haro objects, expanding HII regions and the interaction of supernovae remnants with molecu?lar clouds. Momentum, mass and energy are imparted to the cloud. A full understanding of the shock process is thus needed if we are to understand the structure of molecular clouds and the impact on star formation. Emission from the near-infrared transitions of molecular hydrogen is commonly excited in these shocks. A major puzzle is that emis?sion is seen at velocities that would collisionally dissociate molecular hydrogen, and this is a central question that this thesis seeks to answer. This is approached observationally by trying to relate the observed emission to shock models. Fairly accurate semi-analytic derivations of the emission spectrum expected from hydrodynamic and magnetohydrodynamic molecular shocks are used to fully explore the parameter space of the initial conditions, without resort to expensive numerical calculations. The emission spectrum is then related to that observed. Most of this work is based on a spectroscopic multi-line study of the near-infrared H2 emission in two sources, the Orion outflow and the supernova remnant IC443. These observations are then compared with those expected from the models. In both sources it is found that planar hydrodynamic jump-type shocks (J) are consistent with the new observations. Whplanar magnetically moderated continuous shocks (C), which have been invoked to explain the emission from the shock in Orion, are not. Neither shock types can explain the intensities of CO rotational lines and the H2 line ratios simultaneously. The high velocities that are observed still present a problem. In IC443 the conclusion is the same but, in addition, the pressure needed to explain the observations is higher than that observed in the supernova remnant. It is suggested that this discrepancy may naturally occur when radiative shocks are driven through a clumpy medium. This approach of using line ratios as shock discriminators is extended by velocity resolved spectroscopy of three highly excited emission lines from Orion. These observa?tions demonstrate that there are no discernible differences in the line ratios with velocity despite the large change in the energies of the upper energy levels involved. It is discussed how this further constrains the shock type and limits the contribution from non-thermal excitation (such as fluorescence). The possible physical processes that could lead to high velocity, shocked molecular hydrogen are then discussed. Models proposed in the past are, it is argued, inadequate. It is then shown that the line ratios observed can be closely matched with non-planar continuous type shocks which occur in a bow shock. The densities and pressures needed are still high. The general conclusions are that previous plane parallel C-shock models invoked to explain the molecular shocks are inconsistent with the observations. The line ratios imply that either J-type shocks, in which the cooling takes a long time compared to the initial heating, or C-type bow shocks which produce a range of temperatures are responsible for the emission. It is finally suggested that C-shocks in gas with a very high magnetic field can produce the high velocity H2 emission observed without dissociating the molecules.