The evolving class of objects responsible for the majority of the cosmic X-ray background (XRB) remains at large nearly three and a half decades after the discovery of the XRB. Surveys of sources selected on the basis of their X-ray properties alone provide an unbiased picture of the X-ray sky, but to date they have not been ideal for the discovery of rare types of X-ray sources at faint fluxes: large-area X-ray surveys have been restricted to bright sources, while deep X-ray surveys have been limited to very small patches of sky. X-ray selection coupled with another selection criterion, e.g., a radio or infrared detection, complements “pure” X-ray surveys by (1) permitting the exploration of large areas of sky to faint flux limits for types of extragalactic X-ray sources not well represented in other surveys, and (2) assisting the location of the optical counterparts of these X-ray sources. Using this approach, I have searched for new components of the XRB among the faintest X-ray sources detected by the Einstein Observatory.