Two Decades of Probabilistic Approach to Liouville Conformal Field Theory

Over the past twenty years, the probabilistic approach to Liouville Conformal Field Theory (LCFT) has undergone remarkable developments, transforming a collection of ideas at the interface of probability, geometry, complex analysis and physics into a coherent mathematical theory. Building on Gaussian Free Fields and Gaussian Multiplicative Chaos, rigorous definitions of correlation functions and partition functions have been established, culminating in the probabilistic derivation of the DOZZ formula and a mathematically complete formulation of the conformal bootstrap for LCFT on Riemann surfaces. This survey aims to provide a synthetic account of these advances, emphasizing both the main achievements and the open problems that continue to drive the field.