To uniformly determine the properties of supernova remnants (SNRs) at high energies, we have developed the first systematic survey at energies from 1 to 100 GeV using data from the Fermi Large Area Telescope. Based on the spatial overlap of sources detected at GeV energies with SNRs known from radio surveys, we classify 30 sources as likely GeV SNRs. We also report 14 marginal associations and 245 flux upper limits. A mock catalog in which the positions of known remnants are scrambled in Galactic longitude, allows us to determine an upper limit of 22% on the number of GeV candidates falsely identified as SNRs. We have also developed a method to estimate spectral and spatial systematic errors arising from the diffuse interstellar emission model, a key component of all Galactic Fermi LAT analyses. By studying remnants uniformly in aggregate, we measure the GeV properties common to these objects and provide a crucial context for the detailed modeling of individual SNRs. Combining our GeV results with multiwavelength (MW) data, including radio, X-ray, and TeV, demonstrates the need for improvements to previously sufficient, simple models describing the GeV and radio emission from these objects. We model the GeV and MW emission from SNRs in aggregate to constrain their maximal contribution to observed Galactic cosmic rays.