Using the Einstein Observatory, we have carried out X-ray observations of 107 quasars and have detected 79. From the analysis of this sample of objects we find a correlation between optical emission and X-ray emission. Our data for radio-loud quasars also show a correlation between radio emission and X-ray emission. For a given optical luminosity, the average X-ray emission of radio-loud quasars is approx.3 times higher than that of ratio-quiet quasars. In addition, our data suggest that the radio of X-ray to optical luminosity is decreasing with increasing redshift and/or optical luminosity. Taking into account the differences in X-ray luminosity between radio-loud and radio-quiet quasars, and between low-redshift and high-redshift quasars, we estimate that approx.30% of the observed X-ray background is contributed by quasars brighter than m/sub B/roughly-equal20, while much of the remainder can be contributed by still fainter quasars. Our data also imply that the optical log N--m/sub B/ relation for quasars cannot be extrapolated much beyond m/sub B/roughly-equal20 with the steep slope used to characterize optical source counts at brighter magnitudes. This situation supports the picture in which luminosity evolution, rather than pure density evolution, describes the quasar behavior as a function of redshift. We briefly discuss themore » observed correlation of X-ray luminosity with radio luminosity in the context of current quasar models.« less