The great progress in high-peak-power laser technology has resulted recently in the production of ps and subps laser pulses of PW powers and relativistic intensities (up to 1021 W/cm2) and has laid the basis for the construction of multi-PW lasers generating ultrarelativistic laser intensities (above 1023 W/cm2). The laser pulses of such extreme parameters make it possible to produce highly collimated beams of electrons or ions of MeV to GeV energies, of short time durations (down to subps) and of enormous currents and current densities, unattainable with conventional accelerators. Such particle beams have a potential to be applied in numerous fields of scientific research as well as in medicine and technology development. This paper is focused on laser-driven generation of fast ion beams and reviews recent progress in this field. The basic concepts and achievements in the generation of intense beams of protons, light ions, and multiply charged heavy ions are presented. Prospects for applications of laser-driven ion beams are briefly discussed.