Single-bubble sonoluminescence occurs when an acoustically trapped and periodically driven gas bubble collapses so strongly that the energy focusing at collapse leads to light emission. Detailed experiments have demonstrated the unique properties of this system: the spectrum of the emitted light tends to peak in the ultraviolet and depends strongly on the type of gas dissolved in the liquid; small amounts of trace noble gases or other impurities can dramatically change the amount of light emission, which is also affected by small changes in other operating parameters (mainly forcing pressure, dissolved gas concentration, and liquid temperature). This article reviews experimental and theoretical efforts to understand this phenomenon. The currently available information favors a description of sonoluminescence caused by adiabatic heating of the bubble at collapse, leading to partial ionization of the gas inside the bubble and to thermal emission such as bremsstrahlung. After a brief historical review, the authors survey the major areas of research: Section II describes the classical theory of bubble dynamics, as developed by Rayleigh, Plesset, Prosperetti, and others, while Sec. III describes research on the gas dynamics inside the bubble. Shock waves inside the bubble do not seem to play a prominent role in the process. Section IV discusses the hydrodynamic and chemical stability of the bubble. Stable single-bubble sonoluminescence requires that the bubble be shape stable and diffusively stable, and, together with an energy focusing condition, this fixes the parameter space where light emission occurs. Section V describes experiments and models addressing the origin of the light emission. The final section presents an overview of what is known, and outlines some directions for future research.