We have modeled the line emission from the v = 0-0 S(O), S(2), and S(3), and the v = 1-0 and v = 2-1 S(1) transitions of molecular hydrogen in clouds exposed to high far-ultraviolet fluxes (i.e., photodissociation regions or PDRs) and in shocks. In particular, we have studied the lowest pure rotational H-2 transitions [the 0-0 S(0) and 0-0 S(1) lines], at 28 and 17 mum, respectively. We find that, in PDRs, the emission comes from warm (T greater than or similar to 100 K) molecular gas, situated at optical depths A(v) greater than or similar to 1, beyond the hot atomic surface layer of the clouds. For FUV fields, G0 = 10(3)-10(5) times the average interstellar field and densities n = 10(3)-10(7) CM-3, the typical line intensities are in the range 10(-6) to 10(-4) ergs s-1 cm-2 sr-1. We compare the predictions for the line intensities from both C-type and J-type shock models. For both nondissociative and dissociative molecular shocks, in the same density range, the line intensities range from 10(-6) to 10(-3) ergs s-1 cm-2 sr-1. In the faster, dissociative shocks (v(s) greater than or similar to 25-30 km s-1), the H-2 line emission arises from reformed molecules, downstream of the shock front. Shocked line emission may be detectable from lower velocity shocks (i.e., v(s) approximately 5 km s-1) than previously observed. Measurements of the relative line ratios may be used to help discriminate between competing models for the shock excitation of molecular gas. Observations of these lines will also provide a powerful probe of the extensive regions of warm molecular gas in the Galaxy. The results are applied to recent observations of the 0-0 S(1) transition in both the PDR and the shocked ps in Orion. The Infrared Satellite Observatory (ISO) should detect the v = 0-0 low-J transitions in numerous active regions.