We explore a variety of models in which SN 1961V, one of the most enigmatic supernovae (SNe) ever observed, was a pulsational pair-instability supernova (PPISN). Successful models reproduce the bolometric light curve of the principal outburst and, in some cases, the emission 1 yr before and several years afterward. All models have helium-rich ejecta, bulk hydrogenic velocities near 2000 km s−1, and total kinetic energies of (4−8) × 1050 erg. Each eventually leaves behind a black hole remnant. Three subclasses of PPISN models are explored, each with two different choices of carbon abundance following helium burning. Carbon is an important parameter because shell carbon burning can weaken the explosion. The three subclasses correspond to situations where SN 1961V and its immediate afterglow were (a) a single event, (b) the first of two or more pulsational events separated by decades or centuries, or (c) the latter stages of a complex explosion that had already been going on for a year or more. For the low-carbon case, the main-sequence mass for SN 1961V’s progenitor would have been 100−115 M ⊙, its pre-SN helium core mass was 45−52 M ⊙, and the final black hole mass was 40−45 M ⊙. For the high-carbon case, these values are increased by roughly 20%−25%. In some PPISN models, a ∼1040 erg s−1 star-like object could still be shining at the site of SN 1961V, but it has more likely been replaced by a massive accreting black hole.