A group G is conjugacy separable if any pair of non-conjugate elements remain non-conjugate in a finite quotient of G. While the original motivation for studying conjugacy separability stems from applications to algorithmic problems in group theory, more recently it has been successfully leveraged to exhibit certain rigidity properties of manifolds. In my talk, I will briefly discuss the applications of conjugacy separability to topology, before discussing the new result that all free-by-cyclic groups - a family closely related to 3-manifold groups - are conjugacy separable. This is joint work with Francois Dahmani, Sam Hughes and Nicholas Touikan.