The acceleration of high-energy ion beams (up to several tens of mega-electron-volts per nucleon) following the interaction of short (t < 1 ps) and intense (I2 > 1018 Wcm-2m-2) laser pulses with solid targets has been one of the most active areas of research in the last few years. The exceptional properties of these beams (high brightness and high spectral cutoff, high directionality and laminarity, and short burst duration) distinguish them from the lower-energy ions accelerated in earlier experiments at moderate laser intensities. In view of these properties, laser-driven ion beams can be employed in a number of groundbreaking applications in the scientific, technological, and medical areas. This paper reviews the main experimental results obtained in this area in recent years, the properties of the accelerated beams, the relevant theoretical and computational models, and the main applications that have been implemented or proposed.