  We study the girth of Cayley graphs of finite classical groups G on random
sets of generators. Our main tool is an essentially best possible bound we
obtain on the probability that a given word w takes the value 1 when evaluated
in G in terms of the length of w, which has additional applications. We also
study the girth of random directed Cayley graphs of symmetric groups, and the
relation between the girth and the diameter of random Cayley graphs of finite
simple groups.
