Abstract. In this paper, we mix two well-known approaches of the fault-tolerance: robustness and stabilization. Robustness is the aptitude of an algorithm to withstand permanent failures such as process crashes. The stabilization is a general technique to design algorithms tolerating transient failures. Using these two approaches, we propose algorithms that tolerate both transient and crash failures. We study two notions of stabilization: the self- and the pseudo- stabilization (pseudo-stabilization is weaker than self-stabilization). We focus on the leader election problem. The goal here is to show the implementability of the robust self- and/or pseudo- stabilizing leader election in various systems with weak reliability and synchrony assumptions. We try to propose, when it is possible, communication-efficient implementations. In this work, we exhibit some assumptions required to obtain robust stabilizing leader election algorithms. Our results show, in particular, that the gap betwe...