Higher energies and higher intensities are the necessary conditions for the success of future accelerators. Higher energies need stronger external electromagnetic fields to guide, focus, and accelerate charged particles, while higher intensities result in source of intense self‐fields. In both cases, particle motion deviates considerably from a plain linear evolution as described by the classical Hill equation of transverse betatron motion. Particle stability becomes an issue: this problem can be properly tackled using tools from the nonlinear theory of dynamical systems. The concept of dynamic aperture for single‐particle motion will be presented underlying links with the fundamental theorems of classical mechanics, such as KAM and Nekhoroshev theorems. Modern numerical techniques to compute the dynamic aperture will be discussed with special emphasis on accuracy analysis. Finally, measurements of particle stability in existing circular accelerators will be reviewed. © 2003 American Institute of Physics