The discovery of a diffuse background of cosmic X-radiation accompanied the birth of the field of X-ray astronomy, yet the exact origin of this radiation remains unexplained more than thirty years hence. Subsequent research has demonstrated that the X-ray background (XRB) arises from the integrated emission of individual sources. The principal weakness of the discrete-source explanation for the XRB, however, is that no known class of luminous extragalactic X-ray sources has demonstrated the properties necessary to account for either the intensity or spectrum of the XRB. This dissertation is directed at finding the new class or classes of X-ray sources that do possess the appropriate properties. The approach is based on the premise that the careful optical study of X-ray sources, perhaps selected in novel ways, will eventually elucidate the origin of the XRB. This search for new components of the XRB has used data from three generations of X-ray observatories. I have constructed a catalog of faint X-ray sources in Einstein Observatory images. From this catalog I have drawn radio -selected and infrared-selected samples and obtained spectra of their optical counterparts. I have also obtained optical spectra of objects from a catalog of infrared sources detected in the ROSAT All-Sky Survey, reported initially to contain normal spiral and star- burst galaxies with extreme X-ray luminosities. To the contrary, I find no evidence for X-ray-luminous normal star-forming galaxies in either the ROSAT or Einstein samples. An ASCA observation of the bright X-ray starburst NGC 3256 indicates that its broad -band X-ray spectrum is much too soft for such starbursts to explain the shape of the XRB spectrum. Two new classes of luminous X-ray sources with demure optical properties have been discovered. The first new class is comprised of early-type galaxies which, despite X-ray luminosities typical for broad-line active galactic nuclei (AGNs), have optical spectra that are devoid of emission lines. I have dubbed the second new class of objects "starburst/Seyfert composite" galaxies: their optical spectra are dominated by the characteristics of a starburst, but again, their X-ray luminosities are comparable to those of AGNs. However, close examination of their spectra reveals very subtle AGN signatures. It is likely that the active nucleus in these objects is heavily obscured. New X-ray observations are scheduled to determine the nature of the X-ray emission from both of these new types of objects and their contribution to the XRB.