Liouville CFT is one of the few non-trivial CFTs for which the path integral can be rigorously defined. Starting from this path integral, we give an intrinsic construction of the conformal blocks of the theory, and make contact with the usual formulation of CFT found in algebraic geometry. The key ingredients are a probabilistic construction of the Virasoro algebra, and the spectral resolution of the Hamiltonian. At the end, I will mention some questions left open, such as modular transformations and curvature properties of the bundle of blocks. Joint and ongoing works with Guillarmou, Kupiainen, Rhodes and Vargas.