  A group $G$ is twisted conjugacy separable if for every automorphism
$\varphi$, distinct $\varphi$-twisted conjugacy classes can be separated in a
finite quotient. Likewise, $G$ is completely twisted conjugacy separable if for
any group $H$ and any two homomorphisms $\varphi,\psi$ from $H$ to $G$,
distinct $(\varphi,\psi)$-twisted conjugacy classes can be separated in a
finite quotient. We study how these properties behave with respect to taking
subgroups, quotients and finite extensions, and compare them to other notions
of separability in groups. Finally, we show that for
polycyclic-by-nilpotent-by-finite groups, being completely twisted conjugacy
separable is equivalent to all quotients being residually finite.
