  We describe the construction of the Brera Multi-scale Wavelet (BMW) HRI
Cluster Survey, a deep sample of serendipitous X-ray selected clusters of
galaxies based on the ROSAT HRI archive. This is the first cluster catalog
exploiting the high angular resolution of this instrument. Cluster candidates
are selected on the basis of their X-ray extension only, a parameter which is
well measured by the BMW wavelet detection algorithm. The survey includes 154
candidates over a total solid angle of ~160 deg2 at 10^{-12}erg s^{-1} cm^{-2}
and ~80 deg^2 at 1.8*10^{-13} erg s^{-1}$ cm^{-2}. At the same time, a fairly
good sky coverage in the faintest flux bins (3-5*10^{-14}erg s^{-1} cm^{-2})
gives this survey the capability to detect a few clusters with z\sim 1-1.2,
depending on evolution. We present the results of extensive Monte Carlo
simulations, providing a complete statistical characterization of the survey
selection function and contamination level. We also present a new estimate of
the surface density of clusters of galaxies down to a flux of 3*10^{-14}erg
s^{-1} cm^{-2}, which is consistent with previous measurements from PSPC-based
samples. Several clusters with redshifts up to z=0.92 have already been
confirmed, either by cross-correlation with existing PSPC surveys or from early
results of an ongoing follow-up campaign. Overall, these results indicate that
the excellent HRI PSF (5 arcsec ~FWHM on axis) more than compensates for the
negative effect of the higher instrumental background on the detection of
high-redshift clusters. In addition, it allows us to detect compact clusters
that could be lost at lower resolution, thus potentially providing an important
new insight into cluster evolution.
