  We report on the first results obtained from a new optical identification
programme of 123 faint X-ray sources with $S$(0.5--2$\,{\rm keV)}>2\times
10^{-14}\,$erg$\,$s$^{-1}\,$cm$^{-2}\,$ serendipitously detected in {\it ROSAT}
PSPC pointed observations. We have spectroscopically identified the optical
counterparts to more than 100 sources in this survey. Although the majority of
the sample (68 objects) are QSOs, we have also identified 12 narrow emission
line galaxies which have extreme X-ray luminosities ($10^{42} < L_{\rm X} <
10^{43.5}\,$erg$\,$s$^{-1}$). Subsequent spectroscopy reveals them to be a
mixture of starburst galaxies and Seyfert 2 galaxies in approximately equal
numbers. Combined with potentially similar objects identified in the {\it
Einstein} Extended Medium Sensitivity Survey, these X-ray-luminous galaxies
exhibit a rate of cosmological evolution, $L_{\rm X} \propto
(1+z)^{2.5\pm1.0}$, consistent with that derived for X-ray QSOs. This
evolution, coupled with the steep slope determined for the faint end of the
X-ray luminosity function ($\Phi(L_{\rm X}) \propto L_{\rm X}^{-1.9}$), implies
that such objects could comprise 15--35 per cent of the soft (1--2$\,$keV)
