ABSTRACT I present detailed imaging Fabry-Perot spectrophotometric observations of the nearby prototype starburst galaxy M82, in order to study the high-velocity outflows observed in such galaxies as a property of the energetic starburst phenomena associated with their nuclei. The high spatial and kinematic resolution of our observations has} allowed us to perform photometric analysis of H-alpha, [N II] and [O III] spectral lines at roughly one hundred thousand positions across the extent of the galaxy. Additional model constraints are provided by broadband and narrowband optical imagery, optical spectropolarimetry, and ROSAT soft x-ray imagery. The observed kinematics of the H-alpha-emitting gas in M82 suggest a bipolar outflow of material along the minor axis at a projected velocity of ~300 km s-1, fueled by the bright nuclear starburst regions in the galaxy's disk. All three spectral lines show double components in the centers of the outflowing lobes, with the H-alpha line split by ~300 km s-1 over a region almost a kiloparsec in size. We argue for a model in which the optical emission is radiated by denser ambient material on the surface of "bubbles" that have been evacuated by a hot wind (~108 K) visible at x-ray wavelengths. Comparison of Monte-Carlo simulations with the observed H-alpha and [N II] kinematics shows that the outflow is confined to a cylinder within 350~pc of the disk, flaring outward in a cone beyond that point. The inner optical line-emitting filaments consist primarily of gas that has been entrained from the disk by the outflow, although ambient material in the halo of M82 may be excited by shocks in the outflow at larger radii. The detailed morphology of the bubbles is complex, but the kinematic structure, bubble geometry, and emission line strengths confirm the major predictions of galactic wind hydrodynamical simulations. A spherical halo around M82 is also observed in emission lines. Intermediate [N II]/H-alpha values and spectropolarimetry observations suggest that this halo emission is not intrinsic, but rather disk radiation that has been scattered by a large spherical distribution of dust. Recent large-scale H I maps show massive streamers of atomic gas connecting M81 and M82--residue of a past interaction. We suggest that the dusty halo of M82 may be associated with this H I gas distribution. A remarkably low [N II]/H-alpha ratio is seen in the region of the outflow, implying that the radiation from the surface of the wind bubbles may be produced, in large part, via photoionization of the gas by the nuclear starburst. A corresponding increase in the [O III]/H-alpha ratio along the outflow is also observed. Nevertheless, the importance of shocks as an excitation mechanism is clear in the detailed spatial correlation of the optical and ROSAT soft x-ray emission, particularly at larger radii. Funding for this research has been provided by the office of the Dean of Natural Sciences at Rice University, the Texas Space Grant Consortium, and the Sigma Xi Research Society. Additional funding was provided by AURA/ STScI (grant GO-4382.01.92A), the National Science Foundation (grant AST 88-18900), the William F. Marlar Foundation, the National Optical Astronomy Observatories (NOAO), and Mr. and Mrs. William Gordon.