Liouville theory was introduced by A. Polyakov in 1981 as the theory governing the conformal factor in the summation over all 2d Riemannian metrics. In recent years it has undergone extensive study in the probability community, and numerous conformal field theory (CFT) predictions have been established at a mathematical level of rigor. In this talk we will first explain how one can use probability theory to rigorously define Liouville theory in the path integral approach. In the second part we will survey the main mathematical achievements of this program, including, proof of the DOZZ formula, the conformal bootstrap, properties of conformal blocks, and integration over moduli space. We will also attempt to highlight novel intuitions and formulas coming from probability.