Martingales associated with path-dependent payoff functions are intrinsically linked to path-dependent PDEs. While this connection is typically established via a functional Itô formula, in this talk we present a semigroup-theoretic framework for the analytic characterization of martingales with path-dependent terminal conditions. Specifically, we show that a measurable adapted process of the form V(t) - ∫_0^t Ψ(s)ds is a martingale if and only if a time-shifted version of V is a mild solution to a final value problem (FVP) involving a path-dependent differential operator. We establish existence and uniqueness of solutions to such FVPs using the concept of evolutionary semigroups on path space. We also discuss the relationship between semigroups on path space, nonlinear expectations and their penalty functions. The talk is based on joint work with David Criens, Robert Denk and Markus Kunze.