  We use the number counts of X-ray selected normal galaxies to explore their
evolution by combining the most recent wide-angle shallow and pencil-beam deep
samples available. The differential X-ray number counts, dN/dS, for early and
late-type normal galaxies are constructed separately and then compared with the
predictions of the local X-ray luminosity function under different evolution
scenarios. The dN/dS of early type galaxies is consistent with no evolution out
to z~0.5. For late-type galaxies our analysis suggests that it is the sources
with X-ray--to--optical flux ratio logfx/fopt>-2 that are evolving the fastest.
Including these systems in the late-type galaxy sample yields evolution of the
form ~(1+z)^{2.7} out to z~0.4. On the contrary late-type sources with
logfx/fopt<-2 are consistent with no evolution. This suggests that the
logfx/fopt>-2 population comprises the most powerful and fast evolving
starbursts at moderate and high-z. We argue that although residual
low-luminosity AGN contamination may bias our results toward stronger
evolution, this is unlikely to modify our main conclusions.
