  We investigate the evolution of the 0.5-2 keV soft X-ray luminosity function
(SXLF) of active galactic nuclei (AGN) using results from ROSAT surveys of
various depth, ranging from shallow large-area ROSAT All-Sky Survey -based
samples to the deepest pointed observation on the Lockman Hole. As previously
found, the SXLF evolves rapidly as a function of redshift up to z\sim 1.5 and
is consistent with remaining constant beyond this redshift.
  We found that a form of the Luminosity-Dependent Density Evolution (LDDE)
gives an excellent fit to the data. Extrapolating one form of the LDDE model
(LDDE1) explains \approx 60% of the estimated soft extragalactic Cosmic X-ray
Background (CXRB). We have also found another representation (LDDE2), which
produces \approx 90% of the CXRB and still gives an excellent fit to the sample
AGNs. These two versions of the LDDE models can be considered two extremes of
the possible extrapolations of the SXLF below the flux limit.
  We have also investigated the evolution of the number density of luminous
QSOs. Unlike these QSOs, the ROSAT-selected QSOs do not show evidence for the
decrease of the number density in z>3. The statistical significance of the
difference is, however, marginal.
