  We compute the number counts of clusters of galaxies, the logN-logS relation,
in several X-ray and submm bands on the basis of the Press-Schechter theory. We
pay particular attention to a set of theoretical models which well reproduce
the ROSAT 0.5-2 keV band logN-logS, and explore possibilities to further
constrain the models from future observations with ASCA and/or at submm bands.
The latter is closely related to the European PLANCK mission and the Japanese
Large Millimeter and Submillimeter Array (LMSA) project. We exhibit that one
can break the degeneracy in an acceptable parameter region on the $\Omega_0 -
\sigma_8$ plane by combining the ROSAT logN-logS and the submm number counts.
Models which reproduce the ROSAT band logN-logS will have $N(>S) \sim (150-300)
(S/10^{-12} erg cm^{-2} s^{-1})^{-1.3}$ str$^{-1}$ at $S > 10^{-12} erg cm^{-2}
s^{-1}$ in the ASCA 2-10 keV band, and $N(>S_\nu) \sim (10^2-10^4) (S_\nu/100
mJy)^{-1.5} str^{-1}$ at $S_\nu > 100 mJy$ in the submm (0.85mm) band. The
amplitude of the logN-logS is very sensitive to the model parameters in the
submm band. We also compute the redshift evolution of the cluster number counts
and compare with that of the X-ray brightest Abell-type clusters. The results,
although still preliminary, point to low density ($\Omega_0\sim 0.3$)
universes. The contribution of clusters to the X-ray and submm background
radiations is shown to be insignificant in any model compatible with the ROSAT
logN-logS.
