We identify a "supernova rate problem:" the measured cosmic core-collapse supernova rate is a factor of ∼2 smaller (with significance ∼2σ) 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.