The experimental results are presented on the in-plane resistivity for Zn-substituted single crystals of YBa2Cu3O7-y, and La2-xSrxCuO4. The primary effect of the substituted Zn on the charge transport is to act as a strong potential scatterer in the unitarity limit, producing a large residual resistivity (p0). There appears a critical change in the Zn-substitution effect as the doped hole density Increases. In the underdoped regime, a superconductor-insulator transition is induced at p0 near the universal value (h/4e^2), and Tc is depressed with increase of p0 following a universal curve, independent of material, doped hole density and of species of impurities. By contrast, these universal behaviors are not seen in the overdoped superconducting regime, where the material remains metallic after the superconductivity is suppressed, suggestive of a coexistence of normal and superfluid in this regime.