  We present the X-ray Luminosity Function (XLF) of the ROSAT Deep Cluster
Survey (RDCS) sample over the redshift range 0.05-0.8. Our results are derived
from a complete flux-limited subsample of 70 galaxy clusters, representing the
brightest half of the total sample, which have been spectroscopically
identified down to the flux limit of 4*10^{-14} erg/cm^2/s (0.5-2.0 keV) and
have been selected via a serendipitous search in ROSAT-PSPC pointed
observations. The redshift baseline is large enough that evolutionary effects
can be studied within the sample. The local XLF (z < 0.25) is found to be in
excellent agreement with previous determinations using the ROSAT All-Sky Survey
data. The XLF at higher redshifts, when combined with the deepest number counts
constructed to date (f>2*10^{-14} arg/cm^2/s), reveal no significant evolution
at least out to z=0.8, over a luminosity range 2*10^{42}-3*10^{44} erg/s in the
[0.5-2 keV] band. These findings extend the study of cluster evolution to the
highest redshifts and the faintest fluxes probed so far in X-ray surveys. They
complement and do not necessarily conflict with those of the Einstein Extended
Medium Sensitivity Survey, leaving the possibility of negative evolution of the
brightest end of the XLF at high redshifts.

Astrophysics