We have systematically investigated how binary interaction affects the presupernova evolution of massive close binaries and the resulting supernova explosions, using a Henyey-type stellar evolution code that we have modified to allow its application to binary stellar evolution calculations. With our modified code, we are able to follow the effects of mass and angular momentum loss from the binary, as well as mass transfer within the binary system. We find that a large number of binary scenarios can be distinguished, depending on the type of binary interaction and the evolutionary stage of the supernova progenitor at the time of the interaction. In general, the structure of a massive star can be affected in three fundamentally different ways: by mass loss, mass accretion, or common-envelope evolution. As a result of mass loss by Roche lobe overflow, stars can lose all of their hydrogen-rich envelopes and become helium stars, which are potential candidates for the progenitors of Type Ib supernovae. If the original masses of the binary components are nearly equal, it is possible (for reasonable assumptions about the mass and angular momentum loss rates) that the primary retains part of its hydrogen-rich envelope. In this case, the supernova progenitor would look like a more or less normal red supergiant, even though it may have lost most of its envelope, and the supernova would resemble a classical Type II-L supernova. In most cases, the system remains bound after the explosion, but it may acquire a substantial orbital eccentricity. The system may subsequently become unbound if the original secondary then evolves to become a second supernova. Mass accretion can also significantly alter the structure of the supernova progenitor, if it takes place after the main-sequence phase of the accreting star. The star may then end its life as a blue supergiant instead of a red supergiant. In this case, the resulting supernova explosion would resemble SN 1987A. At the time of the supernova explosion, the presupernova star still has a stellar or (more likely) a neutron star companion. However, since more than half of the total mass of the system is ejected in the supernova explosion, the system is likely to become unbound after the explosion. In the most dramatic case of binary interaction, in which a supernova progenitor captures its companion in a common envelope, two different outcomes are possible, depending on whether the envelope is ejected during the spiral-in phase or remains bound. If the envelope is ejected, the progenitor will become a helium star and the subsequent supernova explosion may be of the Type Ib variety. If the binary components merge completely, the final outcome would be a single star with no trace (except for possible chemical anomalies) of the original secondary. If, during the merger, a significant amount of mass is added to the envelope, the final star may again be a blue supergiant (similar to the results of the accretion scenario), and the resulting supernova would belong to the same class as SN 1987A. In order to assess the importance of the various scenarios, we performed Monte Carlo simulations to estimate the frequencies of occurrence of the individual scenarios. We find that, because of a previous binary interaction, 15%-30% of all massive stars (with initial masses greater than or similar to 8 M.) become helium stars, and another approximately 5% of all massive stars end their lives as blue supergiants rather than as red supergiants. These results may be directly applicable to Type Ib supernovae. Our estimate for the frequency of helium stars is comparable to the observed frequency of Type Ib supernovae, and hence we expect that the explosions of helium stars in binaries account for a substantial fraction of all Type Ib events. Our calculations may also help to answer one of the major puzzles about SN 1987A, namely, the question of why the apparent progenitor (Sk -69-degrees 202) was a blue supergiant rather than a red one, as had been generally expected for the precursors of Type II supernovae. In addition, binary models for SN 1987A may provide plausible explanations for a variety of other anomalies of SN 1987A, ranging from the asymmetric expansion of the ejecta and the variability of the soft X-ray flux to the barium anomaly and the "mystery spot."