The observation of Bose-Einstein condensation in a dilute gas of rubidium in 1995 spawned the new field of ultra-cold, degenerate quantum gases. Unprecedented developments in experimental design and precision control have led to quantum gases becoming the preferred playground for designer quantum many-body systems. This self-contained volume provides a broad overview of the principal theoretical approaches applied to non-equilibrium and finite temperature quantum gases. It covers Bose-Einstein condensates, degenerate Fermi gases, and the more recently realised exciton-polariton condensates in solid state systems. This book fills a gap by providing links between different approaches which have their origins in condensed matter physics, quantum field theory, quantum optics, atomic physics and statistical mechanics. The chapters describing different methodologies have been authored by key researchers in their development, and are organised according to their basic assumptions. The volume uses a unified notation, contains numerous introductory chapters, and is accompanied by editorial notes to guide the reader, providing the first integrated and comparative view of the benefits and shortcomings of the individual approaches. This book is aimed at both graduate students and established researchers wishing to understand the state of the art in non-equilibrium and finite temperature techniques in the exciting and expanding field of quantum gases and liquids. This book should be the first reference point for learning about various theoretical approaches to describing quantum gases. The editors and contributors have created a unique book with well-written articles, meaningful comparisons of various approximation schemes, a uniform notation and more than one thousand references. In addition, the book features introductory chapters and up-to-date review articles of experimental methods and current frontiers. The completeness and depth of the presentation are impressive. -- Wolfgang Ketterle "MIT-Harvard Center for Ultracold Atoms & Nobel Laureate"