The experimental and theoretical evidence for intrinsic reflection-asymmetric shapes in nuclei is reviewed. The theoretical methods discussed cover a wide spectrum, from mean-field theory and its extensions to algebraic and cluster approaches. The experimental data for nuclear ground states and at low and high spin, cited as evidence for reflection asymmetry, are collected and categorized. The extensive data on electric dipole transition moments and their theoretical interpretation are surveyed, along with available data on electric octupole moments. The evidence for reflection-asymmetric molecular states in light nuclei is summarized. The application of reflection-asymmetric theories to descriptions of the fission barrier, bimodal fission, superdeformation, and hyperdeformations is reviewed, and some other perspectives in the wider context of nuclear physics are also given. [S0034-6861(96)00102-X]

DOI:https://doi.org/10.1103/RevModPhys.68.349