A finitely generated residually finite group G is called profinitely rigid, if for any other finitely generated residually finite group H, whenever the profinite completions of H and G are isomorphic, then H is isomorphic to G. In this talk we will review what is known about this in the context of groups arising in low-dimensional geometry and topology. We will then discuss some recent work that constructs finitely presented groups that are profinitely rigid amongst finitely presented groups but not amongst finitely generated one.