By using off-lattice Monte Carlo simulations, the conditions of adsorption of a uniformly charged polyelectrolyte onto oppositely charged planar and spherical surfaces have been investigated. These conditions are functions of the strength of the electrostatic interaction, Debye screening length, chain length, and charge density and curvature of the surface. The adsorption can be tuned by using any one of these parameters. The chain＇s conformation, adsorption energy and thickness of the adsorbed polymer are obtained under different adsorption conditions. We find the Monte Carlo simulation data to be in good agreement with the theoretical prediction derived previously by using the assumptions of ground state dominance and separability.