We present the final results of a ROSAT PSPC program to study the soft X-ray emission properties of a complete sample of low-z quasars. This sample includes all 23 quasars from the Bright Quasar Survey with z ≤ 0.400 and N1.9×1020 cm-2. Pointed ROSAT PSPC observations were made for all quasars, yielding high signal-to-noise (S/N) spectra for most objects, which allowed an accurate determination of the spectral shape. The following main results were obtained: 1. The spectra of 22 of the 23 quasars are consistent, to within ~30%, with a single power-law model at rest-frame 0.2-2 keV. There is no evidence for significant soft excess emission with respect to the best-fit power law. We place a limit (95% confidence) of ~5 × 1019 cm-2 on the amount of excess foreground absorption by cold gas for most of our quasars. The limits are ~1 × 1019 cm-2 in the two highest S/N spectra. 2. The mean 0.2-2 keV continuum of quasars agrees remarkably well with an extrapolation of the mean 1050-350 ? continuum recently determined by Zheng et al. (1996) for z 0.33 quasars. This suggests that there is no steep soft component below 0.2 keV. 3. Significant X-ray absorption (τ 0.3) by partially ionized gas ("warm absorber") in quasars is rather rare, occurring for 5% of the population, which is in sharp contrast to lower luminosity active galactic nuclei (AGNs), where significant absorption probably occurs for ~50% of the population. 4. Extensive correlation analysis of the X-ray continuum emission parameters with optical emission-line parameters indicates that the strongest correlation is between the spectral slope αx and the Hβ FWHM. A possible explanation for this remarkably strong correlation is a dependence of αx on L/LEdd, as seen in Galactic black hole candidates. 5. The strong correlations between αx and L[O III], Fe II/Hβ, and the peak [O III] to Hβ flux ratio are verified. The physical origin of these correlations is still not understood. 6. There appears to be a distinct class of "X-ray-weak" quasars, which form ~10% of the population (three out of 23), where the X-ray emission is smaller, by a factor of 10-30, than expected based on their luminosity at other bands and on their Hβ luminosity. These may be quasars in which the direct X-ray source is obscured and only scattered X-rays are observed. 7. Thin accretion disk models cannot reproduce the observed 0.2-2 keV spectral shape, and they also cannot reproduce the tight correlation between the optical and soft X-ray emission. An as yet unknown physical mechanism must be maintaining a strong correlation between the optical and soft X-ray emission. 8. The H I/He I ratio in the high Galactic latitude ISM must be within 20%, and possibly within 5%, of the total H/He ratio of 10, which indicates that He in the diffuse H II gas component of the interstellar medium is mostly ionized to He II or He III. We finally note the intriguing possibility that although αx in radio-loud quasars (-1.15 ± 0.14) is significantly flatter than in radio-quiet quasars (-1.72 ± 0.09) the X-ray emission may not be related to the presence of radio emission. The difference in αx may result from the strong αx versus Hβ FWHM correlation and the tendency of radio-loud quasars to have broader Hβ.
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