Fundamental properties of hybrid automata, such as existence and uniqueness of executions, are studied. Particular attention is devoted to Zeno hybrid automata, which are hybrid automata that take infinitely many discrete transitions in finite time. It is shown that regularization techniques can be used to extend the Zeno executions of these automata to times beyond the Zeno time. Different types of regularization may, however, lead to different extensions. A water tank control problem and a bouncing ball system are used to illustrate the results.