Addressing graduate students and researchers in physics and mathematics, this book fills a gap in the literature. It is an introduction into modern constructive physics, field theory and statistical mechanics and a survey on the most recent research in this field. It presents the main technical tools such as cluster expansion and their implementation in the rigorous renormalization group, and studies physical models in some detail. The reader will find a study of the ultraviolet limit of the Gross—Neveu model, of continuous symmetry breaking and of self—avoiding random walks in statistical mechanics, as well as applications to solid—state physics. Mathematicians will find constructive methods useful for studies in partial differential equations.