Let G(n) be the free group of finite rank n in a variety V. Since 1982, it has been known that Aut G(n), the automorphism group of G(n), may not be finitely generated for certain n. But is it always true that Aut G(n) is finitely generated for all but a few number of dimensions n? Might this be the case if we restrict the variety V to be locally solvable? Until now, no examples to the contrary were known, and if we require G(n) to be torsion free for almost all n, the questions remain unanswered. Below we shall exhibit varieties for which Aut G(n) is infinitely generated for every n 2: 2. In order to place these results in context and discuss certain unsolved problems, we first discuss the state of knowledge concerning the automorphism groups.