  We identify a "supernova rate problem": the measured cosmic core-collapse
supernova rate is a factor of ~ 2 smaller (with significance ~ 2 sigma) than
that predicted from the measured cosmic massive-star formation rate. The
comparison is critical for topics from galaxy evolution and enrichment to the
abundance of neutron stars and black holes. We systematically explore possible
resolutions. The accuracy and precision of the star formation rate data and
conversion to the supernova rate are well supported, and proposed changes would
have far-reaching consequences. The dominant effect is likely that many
supernovae are missed because they are either optically dim (low-luminosity) or
dark, whether intrinsically or due to obscuration. We investigate supernovae
too dim to have been discovered in cosmic surveys by a detailed study of all
supernova discoveries in the local volume. If possible supernova impostors are
included, then dim supernovae are common enough by fraction to solve the
supernova rate problem. If they are not included, then the rate of dark core
collapses is likely substantial. Other alternatives are that there are
surprising changes in our understanding of star formation or supernova rates,
including that supernovae form differently in small galaxies than in normal
galaxies. These possibilities can be distinguished by upcoming supernova
surveys, star formation measurements, searches for disappearing massive stars,
and measurements of supernova neutrinos.
