We introduce a new, very deep neutral hydrogen (Hi) survey being performed with the Westerbork Synthesis Radio Telescope (WSRT). The Westerbork Hydrogen Accretion in LOcal GAlaxieS (HALOGAS) Survey is producing an archive of some of the most sensitive HI observations available, on the angular scales which are most useful for studying faint, diffuse gas in and around nearby galaxies. The survey data are being used to perform careful modeling of the galaxies, characterizing their gas content, morphology, and kinematics, with the primary goal of revealing the global characteristics of cold gas accretion onto spiral galaxies in the local Universe. In this paper, we describe the survey sample selection, the data acquisition, reduction, and analy...