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 \sim 160 deg ^ { 2 } at 10 ^ { -12 } erg s ^ { -1 } cm ^ { -2 } and \sim 80 deg ^ { 2 } at 1.8 \times 10 ^ { -13 } erg s ^ { -1 } cm ^ { -2 } . At the same time , a fairly good sky coverage in the faintest flux bins ( 3 - 5 \times 10 ^ { -14 } erg s ^ { -1 } cm ^ { -2 } ) gives this survey the capability of detecting 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 \times 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″Â 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 .