  In the paper, we review the recent construction of the Liouville conformal
field theory (CFT) from probabilistic methods, and the formalization of the
conformal bootstrap. This model has offered a fruitful playground to unify the
probabilistic construction of the path integral, the geometric axiomatics of
CFT by Segal and the representation theoretical content of the conformal
bootstrap. We explain and extract the main steps and ideas behind the
construction and resolution of this non-compact CFT.
