E. Artin showed that the braid groups $B_n, n \ge 2$ have faithful representations in $Aut(F_n)$, the automorphism group of the free group. It is well-known that $F_n$ has (uncountably many) bi-orderings: strict total orderings which are invariant under right and left multiplication. We study which braids produce automorphisms which preserve such an ordering and which do not. This corresponds to whether or not certain links in $S^3$ have bi-orderable link group. A sample application is that, of the two hyperbolic 3-manifolds with two cusps and minimal volume, one of them has bi-orderable fundamental group and the other does not. This is joint work with Eiko Kin of Osaka University.