To what extent is a finitely presented group determined by its finite quotients? This question leads naturally to the profinite completion, which encodes all finite quotients of a group in a single algebraic object. In this talk, we will discuss work by Bridson, McReynolds, Reid, and Spitler who construct new examples of groups that are profinitely rigid, meaning they are uniquely determined by their profinite completion, via examples from hyperbolic geometry and low-dimensional topology. Their approach uses techniques involving representation rigidity, linking the finite quotients of a group to arithmetic invariants of lattices in PSL(2,C). If time permits, we will discuss more recent developments in this area pertaining to Dehn fillings of knots.