We have measured the correlation functions of the optically selected clusters of galaxies in the Abell and the APM catalogs, and of the X-ray clusters in the X-ray-Brightest Abell-type Clusters of galaxies (XBACs) catalog and the Brightest Clusters Sample (BCS). The same analysis method and the same method of characterizing the resulting correlation functions are applied to all observational samples. We have found that the amplitude of the correlation function of the APM clusters is much higher than what has been previously claimed, in particular for richer subsamples. The correlation length of the APM clusters with the richness R ≥ 70 (as defined by the APM team) is found to be r0 = 25.4 -3.0 +3.1 h-1 Mpc. The amplitude of correlation function is about 2.4 times higher than that of Croft et al. (1997). The correlation lengths of the Abell clusters with the richness class RC ≥ 0 and 1 are measured to be r0 = 17.4 -1.1 +1.2 and 21.0 -2.8 +2.8 h-1 Mpc, respectively, which is consistent with our results for the APM sample at the similar level of richness. The richness dependence of cluster correlations is found to be r0= 0.40dc + 3.2 where dc is the mean intercluster separation. This is identical in slope with the Bahcall & West (1992)'s estimate, but is inconsistent with the weak dependence of Croft et al. (1997). The X-ray bright Abell clusters in the XBACs catalog and the X-ray selected clusters in the BCS catalog show strong clustering. The correlation length of the XBACs clusters with Lx ≥ 0.65×10 44 h-2 erg s-1 is 30.3 -6.5 +8.2 h-1 Mpc, and that of the BCS clusters with Lx ≥ 0.70×10 44 h-2erg s-1 is 30.2 -8.9 +9.8 h-1 Mpc. The clustering strength of the X-ray clusters is much weaker than what is expected from the optical clusters.