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Providing processes with an eventual leader service is an important issue when one has to design and implement a middleware layer on top of a failure-prone asynchronous distributed system. This invited lecture investigates this problem. It first shows that such a service cannot be built if the underlying system is fully asynchronous. Then, the paper visits several additional behavioral assumptions that have been proposed in the literature to cope with this impossibility and presents corresponding eventual leader election protocols. This lecture can be seen as a guided tour of the eventual leader service problem, whose aim is to benefit researchers and system engineers working in distributed middleware built on top of asynchronous networks.