  We construct the $\Phi^4_3$ measure on an arbitrary 3-dimensional compact
Riemannian manifold without boundary as an invariant probability measure of a
singular stochastic partial differential equation. Proving the nontriviality
and the covariance under Riemannian isometries of that measure gives for the
first time a non-perturbative, non-topological interacting Euclidean quantum
field theory on curved spaces in dimension 3. This answers a longstanding open
problem of constructive quantum field theory on curved 3 dimensional
backgrounds. To control analytically several Feynman diagrams appearing in the
construction of a number of random fields, we introduce a novel approach of
renormalization using microlocal and harmonic analysis. This allows to obtain a
renormalized equation which involves some universal constants independent of
the manifold. We also define a new vectorial Cole-Hopf transform which allows
to deal with the vectorial $\Phi^4_3$ model where $\Phi$ is now a bundle valued
random field. In a companion paper, we develop in a self-contained way all the
tools from paradifferential and microlocal analysis that we use to build in our
manifold setting a number of analytic and probabilistic objects.
