We initiate research on self-stabilization in highly dynamic message-passing systems by addressing the self-stabilizing leader election problem in three wide classes of time-varying graphs (TVGs): the class of TVGs with temporal diameter bounded by Delta (boundDiam), the class of TVGs with temporal diameter quasi-bounded by Delta (quasiBoundDiam), and the class of TVGs with recurrent connectivity only (recur). We first study conditions under which our problem can be solved. We introduce the notion of size-ambiguity to show that the assumption on the knowledge of the number $n$ of processes is crucial. Our results show that any deterministic self-stabilizing leader election algorithm working in the class quasiBoundDiam or recur cannot be size-ambiguous, justifying then the necessity of assuming the exact knowledge of network size in those classes. We then present three self-stabilizing leader election algorithms for Classes boundDiam, quasiBoundDiam, and recur, respectively. Our algorithm for boundDiam stabilizes in at most 2.Delta rounds. In quasiBoundDiam and recur, stabilization time cannot be bounded. However, we show that our solutions are speculative in the sense that their stabilization time in boundDiam is O(Delta) rounds.