We construct the $Φ^4_3$ 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 a non-perturbative, non-topological interacting Euclidean quantum field theory on curved spaces in dimension 3. To control analytically several Feynman diagrams appearing in the construction of a number of random fields, we introduce a novel approach of renormalisation using microlocal and harmonic analysis. This allows to obtain a renormalized equation which involves some universal constants independent of the manifold. 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.