  We establish results concerning the profinite completions of 3-manifold
groups. In particular, we prove that the complement of the figure-eight knot
$S^3-K$ is distinguished from all other compact 3-manifolds by the set of
finite quotients of its fundamental group. In addition, we show that if $M$ is
a compact 3-manifold with $b_1(M)=1$, and $\pi_1(M)$ has the same finite
quotients as a free-by-cyclic group $F_r\rtimes\mathbb{Z}$, then $M$ has
non-empty boundary, fibres over the circle with compact fibre, and
$\pi_1(M)\cong F_r\rtimes_\psi\mathbb{Z}$ for some $\psi\in{\rm{Out}}(F_r)$.
