We present results of the largest, most comprehensive study ever done of the stellar multiplicity of the most common stars in the Galaxy, the red dwarfs. We have conducted an all-sky volume-limited survey for stellar companions to 1120 M dwarf primaries known to lie within 25 pc of the Sun via trigonometric parallaxes. In addition to a comprehensive literature search, stars were explored in new surveys for companions at separations of 2 ''-300 ''. A reconnaissance of wide companions to separations of 300 '' was done via blinking archival images. I-band images were used to search our sample for companions at separations of 2 ''-180 ''. Various astrometric and photometric methods were used to probe the inner 2 '' to reveal close companions. We report the discovery of 20 new companions and identify 56 candidate multiple systems. We find a stellar multiplicity rate of 26.8 +/- 1.4% and a stellar companion rate of 32.4 +/- 1.4% for M dwarfs. There is a broad peak in the separation distribution of the companions at 4-20 au, with a weak trend of smaller projected linear separations for lower mass primaries. A hint that M-dwarf multiplicity may be a function of tangential velocity is found, with faster moving, presumably older, stars found to be multiple somewhat less often. We calculate that stellar companions make up at least 17% of mass attributed to M dwarfs in the solar neighborhood, with roughly 11% of M-dwarf mass hidden as unresolved companions. Finally, when considering all M-dwarf primaries and companions, we find that the mass distribution for M dwarfs increases to the end of the stellar main sequence.