Radiation Pressure Acceleration (RPA) by superintense laser pulses is a promising regime for the acceleration of ions (or matter in general) towards relativistic velocities or to enhance selected properties of ion beams (e.g. large density, high collimation). For linear polarization of the laser pulse, simulations have predicted the dominance of RPA over the well known Target Normal Sheath Acceleration mechanism for very high intensities (>10^22 W/cm^2). For circular polarization (CP) and normal incidence on target, RPA dominates at any intensity due to the suppression of fast electron generation. Several theoretical groups have thus proposed RPA with CP and ultrathin targets as a route towards highly efficient ion acceleration towards very high energies. In this talk we will review basic models of RPA and show the results of particle-in-cell simulations in various dimensionality aimed at clarifying relevant issues of RPA. These simulations will include a parametric study of ultrathin target acceleration in 1D to find optimal values of the target thickness, a 2D study of surface rippling and its dependence on laser polarization, and 3D simulations addressing the issue of angular momentum absorption and magnetic field generation.