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The term stochastic hybrid systems defines a class of dynamical and control systems that involve the interaction of continuous dynamics, discrete dynamics and probabilistic uncertainty. Over the last decade stochastic hybrid systems have emerged as a powerful modeling paradigm in a wide range of application areas. The talk will provide an overview of recent developments in this rapidly evolving field of research. We will introduce some of the theoretical foundations and challenges of stochastic hybrid systems and outline computational methods that can be used to analyze and control such systems, based primarily on randomized algorithms. The discussion will be motivated by several applied modeling, analysis and control problems from the areas of systems biology and air traffic management. This extended abstract provides some motivation for the talk, as well as an extensive list of references that can serve as a starting point for further reading.