The acceleration of ions in the interaction of high intensity laser pulses with overdense plasmas is investigated with particle-in-cell simulations. For circular polarization of the laser pulses, highdensity ion bunches moving into the plasma are generated at the laser-plasma interaction surface. A simple analytical model accounts for the numerical observations and provides scaling laws for the ion bunch energy and generation time as a function of pulse intensity and plasma density.PACS numbers: 52.38.-r, 52.38. Kd, 52.50.Jm, 52.65.Rr The study of the interactions between ultra-intense laser pulses and plasmas has proved to be a very rich soil where technological progress and fundamental physics meet each other. Particularly intriguing is the concept of laser-plasma based ion acceleration. From astrophysics [1], to medical hadrontherapy [2], from proton radiography [3], to nuclear physics [4], from proton imaging techniques [5], to nuclear fusion [6], the problem of accelerating and manipulating charged particles with laserplasma interactions offers a series of challenges ranging from fundamental to applied physics, thus a clear understanding of the basic mechanisms is mandatory. Several recent experiments have reported the emission of energetic ions from solid targets [7]. It is still a matter of debate whether the ions are mainly accelerated at the rear surface of the target (by the field generated by fast electrons escaping in vacuum [8]) or at the front surface involving phenomena such as acceleration by a collisionless electrostatic shock [9,10,11], by a solitary wave [12] or by ion trapping in a propagating double layer [13].In this work we elucidate an even more basic process of ion acceleration in cold plasmas, purely related to the formation of an electrostatic field due to the action of the laser ponderomotive force on the electrons and, consequently, on the ions via space charge displacement. This investigation shows both the necessity of a kinetic description of this process and the fundamental role played by the laser light polarization by showing the differences between circular and linear one. It will be shown by particle-in-cell (PIC) simulations that circularly polarized light gives rise to a "pulsed"acceleration and produces ion bunches directed into the target. A simple analytical model is used to explain the acceleration dynamics and for the deduction of scaling laws that relate the interaction parameters to the energy acquired by the ions. With respect to other concepts for laser ion acceleration, the present mechanism with circularly polarized light leads to very high densities in the bunches, as might be of interest for problems of compression and acceleration of high-density matter.We consider a laser pulse impinging on a cold, step- is the critical density for a laser with carrier frequency ω L , ω p is the plasma frequency and m e , e are the electron mass and charge. The laser field amplitude will be given in units of the dimensionless parameter a L = (eE L /m e ω L c). In the PIC s...