  We calculate the all-sky number of galaxy clusters that are expected to be
gravitationally lensed by foreground massive clusters. We describe the redshift
and number distributions of clusters using a Press-Schechter analysis, and
model the foreground lensing clusters as singular isothermal spheres. If
Omega_m=0.3 and Omega_Lambda=0.7, we expect ~ 30 cluster-cluster strong lensing
events that involve foreground X-ray luminous clusters with total mass greater
than 7.5 x 10^14 h^-1 M_sun, or X-ray luminosity L_x (2-10 keV) 8 x 10^44 h^-2
ergs s^-1, and background clusters with total mass greater than 10^14 h^-1
M_sun. The number expected in an open universe with Omega_m = 0.3 is less than
\~ 4. Because of uncertainty in sigma_8, the root-mean-square density
fluctuations in spheres of radius 8 h^-1 Mpc, the exact number of such lensing
events is uncertain by a factor of about 5. We examine methods to detect
cluster-cluster lensing events based on optical, X-ray, and Sunyaev-Zel'dovich
effect observations.
