Two-dimensional Liouville conformal field theory has been a subject of much investigation since its first appearance in the study of non-critical string theory. We consider a generalization of the two-dimensional Liouville CFT to any number of even dimensions. We also construct a four supercharges N=1 Liouville superconformal field theory in four dimensions. The theories contain log-correlated scaler fields with a background Q-curvature charge and an exponential potential. The theories are non-unitary with a continuous spectrum of scaling dimensions. We study the classical solutions, the conformal anomalies and the correlation functions. The presented results are based on two papers.