Higher order statistics are investigated in Ω cold dark matter (CDM) universes by analyzing 500 h-1 Mpc high-resolution tree N-body simulations with both Ω=1 and Ω<1. The amplitudes of the N-point correlation functions are calculated from moments of counts-in-cells determined by a pair of new algorithms especially developed for large simulations. This approach enables massive oversampling with 109-1014 cells for accurate determination of factorial moments from up to 47 million particles in the scale range of 8 h-1 kpc-125 h-1 Mpc. Thorough investigation shows that there are three scale ranges in the simulations: ≥8 h-1 Mpc, a weakly nonlinear regime where perturbation theory applies with utmost precision; 1-8 h-1 Mpc, the nonlinear plateau; and finally ≤1 h-1 Mpc, a regime where dynamical discreteness effects dominate the higher order statistics. In the physically relevant range of 1-125 h-1 Mpc the results (1) confirm the validity of perturbation theory in the weakly nonlinear regime; (2) establish the existence of a plateau in the highly nonlinear regime similar to the one observed in scale-free simulations; (3) show extended perturbation theory to be an excellent approximation for the nonlinear regime; (4) find the time-dependence of the SN's to be negligible in both regimes; (5) in comparison with similar measurements in the Edinburgh-Durham Southern Galaxy Catalog survey, strongly support Ω<1 with no biasing; and (6) show that the formulae of Szapudi and Colombi provide a good approximation for errors on higher order statistics measured in N-body simulations.