This paper presents new model results describing the evolution of massive close binaries from their initial ZAMS to post-supernova stages. Unlike the previous conservative study by Stone (Astrophys. J. 232, 520 (1979) (Paper II)), these results allow explicitly for mass loss from the binary system occurring during the core hydrogen- and helium-burning stages of the primary binary star as well as during the Roche lobe overflow. Because of uncertainties in these rates, model results are given for several reasonable choices for these rates. All of the models consistently predict an increasing relation between the peculiar space velocities and masses for runaway OB stars which agrees well with the observed correlations discussed in Stone (Astron. J. 86, 544 (1981) (Paper III)) and also predict a lower limit at Mroughly-equal11M/sub sun/ for the masses of runaway stars, in agreement with the observational limit found by A. Blaauw (Bull. Astron. Inst. Neth. 15, 265, 1961), both of which support the binary-supernova scenario described by van den Heuvel and Heise for the origin of runaway stars. These models also predict that the more massive O stars will produce correspondingly more massive compact remnants, and that most binaries experiencing supernova-induced kick velocities of magnitude V/submore » k/> or approx. =300 km s/sup -1/ will disrupt following the explosions. The best estimate for this velocity as established from pulsar observations is V/sub k/roughly-equal150 km s/sup -1/, in which case probably only 15% if these binaries will be disrupted by the supernova explosions, and therefore, almost all runaway stars should have either neutron star or black hole companions.« less