  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 one year before and several
years afterward. All models have helium-rich ejecta, bulk hydrogenic velocities
near 2000 km s$^{-1}$, and total kinetic energies from 4 to 8 $\times 10^{50}$
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 to 115 \Msun; its pre-SN helium core mass was 45
to 52 \Msun; and the final black hole mass, 40 to 45 \Msun. For the high-carbon
case, these values are increased by roughly 20 to 25\%. In some PPISN models, a
$\sim10^{40}$ 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of SN~1961V, but it has more likely been replaced by a massive accreting black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.
