  Initiated by Polyakov in his 1981 seminal work, the study of two-dimensional
Liouville Conformal Field Theory has drawn considerable attention over the past
decades. Recent progress in the understanding of conformal geometry in
dimension higher than two have naturally led to a generalization of Polyakov
formalism to higher dimensions, based on conformally invariant operators:
Graham-Jenne-Mason-Sparling operators and the $\mathcal{Q}$-curvature.
  This document is dedicated to providing a rigorous construction of Liouville
Conformal Field Theory on even-dimensional spheres. This is done at the
classical level in terms of a generalized \textit{Uniformization} problem, and
at the quantum level thanks to a probabilistic construction based on
log-correlated fields and Gaussian Multiplicative Chaos. The properties of the
objects thus defined are in agreement with the ones expected in the physics
literature.
