The fundamental lemma and stabilization

Motivated by the Langlands functoriality conjecture, we would like to stabilize the (twisted) trace formula following Langlands, Kottwitz, Arthur, Clozel, Labesse, and Moeglin-Waldspurger. The goal of this expository talk is to make sense of this, in some detail as far as regular elliptic terms are concerned, underlining the role played by the fundamental lemma and its variants. An overview will be given of the relationship between various versions of the fundamental lemma and the transfer conjecture (twisted, weighted, Lie algebra version, equal-characteristic).