Large surveys of galaxy clusters with the Hubble Space Telescope (HST) and Spitzer, including the Cluster Lensing And Supernova survey with Hubble and the Frontier Fields, have demonstrated the power of strong gravitational lensing to efficiently deliver large samples of high-redshift galaxies. We extend this strategy through a wider, shallower survey named RELICS, the Reionization Lensing Cluster Survey, described here. Our 188-orbit Hubble Treasury Program observed 41 clusters at 0.182<=z<=0.972 with Advanced Camera for Surveys (ACS) and WFC3/IR imaging spanning 0.4-1.7{mu}m. We selected 21 of the most massive clusters known based on Planck PSZ2 estimates and 20 additional clusters based on observed or inferred lensing strength. RELICS observed 46 WFC3/IR pointings (~200arcmin^2^) each with two orbits divided among four filters (F105W, F125W, F140W, and F160W) and ACS imaging as needed to achieve single-orbit depth in each of three filters (F435W, F606W, and F814W). As previously reported by Salmon+ (2020ApJ...889..189S), we discovered over 300 z~6-10 candidates, including the brightest z~6 candidates known, and the most distant spatially resolved lensed arc known at z~10. Spitzer IRAC imaging (945hr awarded, plus 100 archival, spanning 3.0-5.0{mu}m) has crucially enabled us to distinguish z~10 candidates from z~2 interlopers. For each cluster, two HST observing epochs were staggered by about a month, enabling us to discover 11 supernovae, including 3 lensed supernovae, which we followed up with 20 orbits from our program.