The question of whether the well-foundedness of cardinals implies the Axiom of Choice has been raised on many occasions, and remains a major open problem. In the complete absence of Choice, however, there turn out to be various notions of well-foundedness. These notions will be discussed, alongside the implications among them. Some implications between these and other weak forms of Choice (including Dependent Choice and the Partition Principle) will also be discussed. This talk is based on joint work with Andreas Blass.