We have demonstrated the promising radiation pressure acceleration (RPA) mechanism of laser-driven ion acceleration at currently achievable laser and target parameters through a large number of two-dimensional particle-in-cell simulations and experiments. High-density monoenergetic ion beams with unprecedented qualities such as narrow-peaked spectrum, lower-divergence and faster energy-scaling are obtained, compared with the conventional target normal sheath acceleration. The key condition for stable RPA from thin foils by intense circularly polarized lasers has been identified, under which the stable RPA regime can be extended from ultrahigh intensities >1022 W cm−2 to a currently accessible range 1020–1021 W cm−2. The dependences of the RPA mechanism on laser polarization, intensity and on the target composition and areal density have been studied.