The purpose of this book is to present a modern account of mapping theory with emphasis on quasiconformal mapping and its generalizations. The modulus method was initiated by Arne Beurling and Lars Ahlfors to study conformal mappings, and later this method was extended and enhanced by several others. The techniques are geometric and they have turned out to be an indispensable tool in the study of quasiconformal and quasiregular mappings as well as their generalizations. The book is based on recent research papers and extends the modulus method beyond the classical applications of the modulus techniques presented in many monographs. TOC:Introduction and notation.- Moduli and capacity.- Moduli and domains.- Q-Homeomorphisms.- Q-Homeomorphisms with Q in BMO.- More General Q-Homeomorphisms.- Ring Q-Homeomorphisms.- Mappings With Finite Length Distortions (FLD).- Lower Q-Homeomorphisms.- Mappings With Finite Area Distortions (FAD).- On Ring Solutions of the Beltrami Equation.- Homeomorphisms with Finite Means Dilatations.- On the Mapping Theory in Metric Spaces.- Moduli Theory.- BMO Functions by John-Nirenberg.- References.- Index.