There has been a growing interest in the study of hybrid dynamical systems around the globe. Such systems arise in many applications such as switching circuits in power electronics, orbital transfer and control of spacecrafts, multimedia switching communication networks, drug administration in cancer chemotherapy, interest rate adjustment, constrained impact mechanics, and ecosystems management. Hybrid dynamical systems provide a natural framework for mathematical modeling of complex reactive systems in which physical processes interact with man-made automated environments. However, there are many challenging issues. Many theorems of continuous dynamical systems fail on hybrid dynamical systems, which is especially so when the systems contain time delay and stochastic disturbance. New ideas need to be developed that deal with impacts, jumps, switching, sliding and chattering problems in a robust and easily applicable way. This talk will discuss some recent results and potential research problems in hybrid dynamical systems.