Abstract: We initiate research on self-stabilization in highly dynamic identified message-passing systems where dynamics is modeled using time-varying graphs (TVGs). More precisely, we address the self-stabilizing leader election problem in three wide classes of TVGs: the class TCB (Δ) of TVGs with temporal diameter bounded by Δ, the class TCB (Δ) of TVGs with temporal diameter quasi-bounded by Δ, and the class TCR of TVGs with recurrent connectivity only, where TCB (Δ) ⊆ TCB (Δ) ⊆ TCR. We first study conditions under which our problem can be solved. Precisely, we introduce the notion of size-ambiguity to show that the assumption on the knowledge of the number n of processes is central. Our results reveal that, despite the existence of unique process identifiers, any deterministic self-stabilizing leader election algorithm working in the TVG class TCB (Δ) or TCR cannot be size-ambiguous, justifying why our solutions for those classes assume the exact knowledge of n. We then present three self-stabilizing leader election algorithms for the TVG classes TCB (Δ), TCB(Δ), and TCR, respectively.