I will survey recent results regarding the dynamics of positive definite functions and character rigidity of irreducible lattices in higher-rank semisimple algebraic groups. These results have several applications to ergodic theory, topological dynamics, unitary representation theory, and operator algebras. In the case of lattices in higher-rank simple algebraic groups, I will explain the key operator algebraic novelty, which is a noncommutative Nevo-Zimmer theorem for actions on von Neumann algebras. I will also present a noncommutative Margulis' factor theorem and discuss its relevance regarding Connes' rigidity conjecture for group von Neumann algebras of higher-rank lattices.