  In 1987 Graeme Segal gave a functorial definition of Conformal Field Theory
(CFT) that was designed to capture the mathematical essence of the Conformal
Bootstrap formalism pioneered in physics by Belavin-Polyakov-Zamolodchikov. In
Segal's formulation the basic objects of CFT, the correlation functions of
conformal primary fields, are viewed as functions on the moduli space of
Riemann surfaces with marked points which behave naturally under gluing of
surfaces. In this paper we give a probabilistic realization of Segal's axioms
in Liouville Conformal Field Theory (LCFT) which is a CFT that plays a
fundamental role in the theory of random surfaces and two dimensional quantum
gravity. Then we use Segal's axioms to express the correlation functions of
LCFT in terms of the basic objects of LCFT: its {\it spectrum} and its {\it
structure constants}, determined in earlier works by the authors. As a
consequence, we obtain a formula for the correlation functions as multiple
integrals over the spectrum of LCFT, the structure of these integrals being
associated to a pant decomposition of the surface. The integrand is the modulus
squared of a function called conformal block: its structure is encoded by the
commutation relations of an algebra of operators called the Virasoro algebra
and it depends holomorphically on the moduli of the surface with marked points.
The integration measure involves a product of structure constants, which have
an explicit expression, the so called DOZZ formula.
