  We present a solution to the Conjugacy Problem in the group of
outer-automorphisms of $F_3$, a free group of rank 3. We distinguish according
to several computable invariants, such as irreducibility, subgroups of
polynomial growth, and subgroups carrying the attracting lamination. We
establish, by considerations on train tracks, that the conjugacy problem is
decidable for the outer-automorphisms of $F_3$ that preserve a given rank 2
free factor. Then we establish, by consideration on mapping tori, that it is
decidable for outer-automorphisms of $F_3$ whose maximal polynomial growth
subgroups are cyclic. This covers all the cases left by the state of the art.
