Liouville Conformal Field Theory (LCFT) is an essential building block of Polyakov’s formulation of non critical string theory. Moreover, scaling limits of statistical mechanics models on random lattices (planar maps) are believed to be described by LCFT as well as partition functions of certain 4-dimensional Yang-Mills theories. I will give an introduction to a rigorous probabilistic construction of LCFT (joint work with F. David, R. Rhodes and V. Vargas) and sketch a proof (jointly with R. Rhodes and V. Vargas) of the remarkable explicit expression of the three point functions of LCFT conjectured by Dorn, Otto, Zamolodchicov and Zamolodchikov.