Let G be a finite simple group and let S be a normal subset of G. We determ
ine the diameter of the Cayley graph F(G, S) associated with G and S, up to
a multiplicative constant. Many applications follow. For example. we deduc
e that there is a constant c such that every element of G is a product of c
involutions (and we generalize this to elements of arbitrary order). We al
so show that for any word w = w (x(1),. . .,x(d)). there is a constant c =
c(w) such that for any simple group G on which w does not vanish, every ele
ment of G is a product of c values of w. From this we deduce that every ver
bal subgroup of a semisimple profinite group is closed. Other applications
concern covering numbers, expanders, and random walks on finite simple grou
ps.