This monograph deals with recent questions of conformal geometry. It provides in detail an approach to studying moduli spaces of conformal structures, using a new canonical metric for conformal structures. This book is accessible to readers with basic knowledge in differential geometry and global analysis. It addresses graduates and researchers.

Preliminaries.- A canonical metric for flat conformal manifolds.- Kleinian groups and moduli spaces.- Asymptotics: The flat case.- Generalization in low dimensions.- The moduli space of all conformal structures.