  We present the results of a systematic search for old isolated neutron stars
(ONSs) in the direction of two giant molecular clouds in Cygnus (Rift and OB7).
From theoretical calculations, we expect the detection of a large number of
ONSs with the PSPC on board ROSAT. By analyzing the PSPC pointings in the
direction of the clouds, we find four sources characterized by count rates
($\sim 10^{-3}$ c s$^{-1}$) and spectral properties consistent with the
hypothesis that the X--ray radiation is produced by ONSs and also characterized
by the absence of any measurable optical counterpart within their error circle
in the digitized red plates of the Palomar All Sky Survey. The importance of
follow--up deep observations in the direction of these ONS candidates is
discussed. The observational and theoretical approach presented here could be
fruitfully applied also to the systematic search for ONSs in other regions of
the Galaxy.
