I will introduce the Furstenberg entropy theory in ergodic group theory. As the main result, I will present a rigidity theorem for Furstenberg entropy: a quasi-factor map between two stationary spaces preserves the Furstenberg entropy if and only if it induces an isomorphism between their Radon-Nikodym factors. I will also discuss applications of this rigidity theorem in establishing rigidity phenomena for Poisson boundaries. This result and its proof are inspired by its counterpart in the noncommutative setting of von Neumann algebras, which I will briefly introduce at the end if time permits.