This dissertation presents a study of shock-excited molecular hydrogen (H2) in the interstellar medium. The aims of this thesis are to understand the shock-excitation process and to understand the global role of shocks in the interstellar medium. These aims are quantified as the investigation of specific problems. To address the problems, a variety of observing techniques have been applied and several sources studied, with particular reference on supernova remnant IC 443. An analytical model for the cooling flow behind a shock has also been developed. The observations show that extensive regions of low surface brightness H2 line emission are common in shocked molecular sources. Models are presented for three sources; IC 443, CRL 616 and OMC-1. In IC 443 emission comes from a sinuous ridge, about 20 parsecs long ans less than a parsec wide, with over 20 bright emission peaks distributed along it. The total H2 line luminosity is ~1600L, making IC 443 one of the brightest galactic H2 emission line objects yet detected. The spatial distributions of accelerated line emissions from other molecules (CO, HCO+, HCN) and atomic gas (HI) are remarkably similar to that of shocked H2. There is evidence for partial dissociation of molecular gas by the shock, but there can be little ionised gas present. Important cooling mechanisms for the hot gas (and possibly the dominant mechanisms) are H2 line radiation and H2 dissociation, except possibly in the densest clumps where far-IR emission from collisionally heated grains may dominate. H" line profiles were obtained in several sources and show considerable variation between sources. In CRL 618 the line is ~250km/s wide, the largest yet measured for a galactic source, and is composed of several discrete components. the high-velocity line emission is interpreted as being due to the shocking of high-velocity, discrete molecular clumps, embedded in and shocked by a stellar wind. Line polarization measurements in OMC-1 show there are two distinct regions of H2 line emission. In the outflow region the line is dichroically polarized by a slab of alligned grains lying between us and the outflow, with polarization vectors parallel to the outflow axis. The alignment mechanism is possibly due to the agency of a magnetic field, and thus the polarization vectors may trace the magnetic field direction which is therefore aligned with the outflow axis. Outside the core region the polarization vectors show a centro-symmetric pattern characteristic of scattering, centred on the region of peak molecular hydrogen emission. This amounts to the discovery of a molecular hydrogen reflection nebula. Observations of five H2 lines, in four types of sources, show no major differences in relative line ratios between sources. This is dispite different pre-shock conditions being expected in each source. The shocked gas cannot be characterised by a single excitation temperature. An analytical model has been developed for the cooling flow behind a jump-shock into molecular gas, driven by an isobaric thermal pressure. The model predicts that, when the density is larger than the critical density needed to thermalise the level populations of the dominant coolant, and t the post-shock temperature is sufficiently large, then the line ratios only depend on the upper-state level energies of the lines and on the form of the cooling function. For the observed excitation temperatures, the dominant cooling mechanism, consistent with the data, is cooling through the vibrational/ rotational lines of the hydrogen molecule itself. This conclusion applies when the temperature is in the range ~500 - 4000 K and the gas density is >10(5)cm-3