Over the past twenty-five years, several striking predictions from statistical physics have been rigorously proven using Schramm–Loewner evolution (SLE). Under fairly general conditions, establishing the convergence of interface curves in critical 2D lattice models to their scaling limits requires only a single martingale observable -- a stochastic analogue of an integral of motion -- which uniquely determines the law. In this talk, I will describe how to construct families of such martingale-observables in various conformal settings, deriving them from Ward’s equations in conformal field theory and variational formulas in geometric analysis. This presentation is based on a series of joint works with Tom Alberts, Sung-Soo Byun, and Nikolai Makarov.