This survey addresses stability analysis for stochastic hybrid systems (SHS), which are dynamical systems that combine continuous change and instantaneous change and that also include random effects. We re-emphasize the common features found in most of the models that have appeared in the literature, which include stochastic switched systems, Markov jump systems, impulsive stochastic systems, switching diffusions, stochastic impulsive systems driven by renewal processes, diffusions driven by Levy processes, piecewise-deterministic Markov processes, general stochastic hybrid systems, and stochastic hybrid inclusions. Then we review many of the stability concepts that have been studied, including Lyapunov stability, Lagrange stability, asymptotic stability, and recurrence. Next, we detail Lyapunov-based sufficient conditions for these properties, and additional relaxations of Lyapunov conditions. Many other aspects of stability theory for SHS, like converse Lyapunov theorems and robustness theory, are not fully developed; hence, we also formulate some open problems to serve as a partial roadmap for the development of the underdeveloped pieces. (C) 2014 Elsevier Ltd. All rights reserved.