  We present the largest compilation to date of optical observations during and
following fast radio bursts (FRBs). The data set includes our dedicated
simultaneous and follow-up observations, as well as serendipitous archival
survey observations, for a sample of 15 well-localized FRBs: eight repeating
and seven one-off sources. Our simultaneous (and nearly simultaneous with a
$0.4$ s delay) optical observations of 13 (1) bursts from the repeating FRB
20220912A provide the deepest such limits to date for any extragalactic FRB,
reaching a luminosity limit of $\nu L_\nu\lesssim 10^{42}$ erg s$^{-1}$
($\lesssim 2\times10^{41}$ erg s$^{-1}$) with $15-400$ s exposures; an
optical-flux-to-radio-fluence ratio of $f_{\rm opt}/F_{\rm radio}\lesssim
10^{-7}$ ms$^{-1}$ ($\lesssim 10^{-8}$ ms$^{-1}$); and flux ratio of $f_{\rm
opt}/f_{\rm radio}\lesssim 0.02-\lesssim 2\times 10^{-5}$ ($\lesssim 10^{-6}$)
on millisecond to second timescales. These simultaneous limits provide useful
constraints in the context of FRB emission models, such as the pulsar
magnetosphere and pulsar nebula models. Interpreting all available optical
limits in the context of the synchrotron maser model, we find that they
constrain the flare energies to $\lesssim 10^{43}-10^{49}$ erg (depending on
the distances of the various repeating FRBs, with $\lesssim 10^{39}$ erg for
the Galactic SGR 1935+2154). These limits are generally at least an order of
magnitude larger than those inferred from the FRBs themselves, although in the
case of FRB 20220912A our simultaneous and rapid follow-up observations
severely restrict the model parameter space. We conclude by exploring the
potential of future simultaneous and rapid-response observations with large
optical telescopes.
