In this talk, I will present an approach to conformally invariant Liouville geometry in dimensions n>2. In particular, I will discuss* conformally invariant log-correlated Gaussian fields based on Paneitz and GJMS operators, * Liouville quantum gravity measures, Liouville Brownian motion on n-dim manifolds,* Polyakov-Liouville measure and conformal anomaly involving Branson’s Q-curvature.