First, I will give introduce and motivate conformal field theory from an analytic point of view. The main message is that we can understand Segal's definition of CFT as a generalisation of the Hille-Yosida theorem. While Segal's axioms are usually stated in an algebraic language, this reformulation gives a valuable entry point to CFT for analysts (and probabilists). Second, I will illustrate the relevance of this reformulation by explaining how Segal's axioms are fully realised in Liouville CFT. Based on joint works with Guillarmou, Kupiainen, Rhodes and Vargas.