This lecture is the third part of a series on Probabilistic Conformal Field Theory presented by Antti Kupiainen at the Hausdorff Center for Mathematics. Explore the mathematical foundations of Conformal Field Theories (CFTs), which describe universal behavior of physical systems at second order phase transition points and are crucial in Quantum Field Theories. Learn about the probabilistic approach to CFT based on path integral formulation and its connection to Graeme Segal's geometric approach to conformal bootstrap. Discover two prominent CFTs: Liouville CFT, central to Liouville Quantum Gravity and random surfaces theory, and Wess-Zumino-Witten models with their rich representation theoretical content. Delve into the mathematical structures of two-dimensional CFTs first uncovered by Belavin, Polyakov, and Zamolodchicov in 1983, which have profound impacts on representation theory and geometry.