We present StrayCats, a catalog of Nuclear Spectroscopic Telescope ARray (NuSTAR) stray light observations of X-ray sources. Stray light observations arise for sources 1{deg}-4{deg} away from the telescope pointing direction. At this off-axis angle, X-rays pass through a gap between the optics and aperture stop and so do not interact with the X-ray optics; instead, they directly illuminate the NuSTAR focal plane. We have systematically identified and examined over 1400 potential observations resulting in a catalog of 436 telescope fields and 78 stray light sources that have been identified. The sources identified include historically known persistently bright X-ray sources, X-ray binaries in outburst, pulsars, and type I X-ray bursters. In this paper, we present an overview of the catalog, how we identified the StrayCats sources, and the analysis techniques required to produce high-level science products. Finally, we present a few brief examples of the science quality of these unique data.