  For large classes of even-dimensional Riemannian manifolds $(M,g)$, we
construct and analyze conformally invariant random fields. These centered
Gaussian fields $h=h_g$, called co-polyharmonic Gaussian fields, are
characterized by their covariance kernels $k$ which exhibit a precise
logarithmic divergence: $|k(x,y)-\log\frac1{d(x,y)}|\le C$. They share a
fundamental quasi-invariance property under conformal transformations.
  In terms of the co-polyharmonic Gaussian field $h$, we define the quantum
Liouville measure, a random measure on $M$, heuristically given as $$
d\mu_g^{h}(x):= e^{\gamma h(x)-\frac{\gamma^2}2k(x,x)}\,d \text{vol}_g(x)$$ and
rigorously obtained as almost sure weak limit of the right-hand side with $h$
replaced by suitable regular approximations $h_\ell, \ell\in{\mathbb N}$. In
terms on the quantum Liouville measure, we define the Liouville Brownian motion
on $M$ and the random GJMS operators.
  Finally, we present an approach to a conformal field theory in arbitrary even
dimensions with an ansatz based on Branson's $Q$-curvature: we give a rigorous
meaning to the Polyakov-Liouville measure $$ d\boldsymbol{\nu}^*_g(h)
=\frac1{Z^*_g} \exp\Big(- \int \Theta\,Q_g h + m e^{\gamma h} d
\text{vol}_g\Big) \exp\Big(-\frac{a_n}{2} {\mathfrak p}_g(h,h)\Big) dh, $$ and
we derive the corresponding conformal anomaly.
  The set of admissible manifolds is conformally invariant. It includes all
compact 2-dimensional Riemannian manifolds, all compact non-negatively curved
Einstein manifolds of even dimension, and large classes of compact hyperbolic
manifolds of even dimension. However, not every compact even-dimensional
Riemannian manifold is admissible.
  Our results rely on new sharp estimates for heat kernels and higher order
Green kernels on arbitrary compact manifolds.
