  We construct explicit generating sets S_n and \tilde S_n of the for the
alternating and the symmetric groups, which turn the Cayley graphs C(Alt(n),
S_n) and C(Sym(n), \tilde S_n) into a family of bounded degree expanders for
all n. This answers affirmatively an old question which has been asked many
times in the literature. These expanders have many applications in the theory
of random walks on groups, card shuffling and other areas.
