This volume presents a detailed account of a new method for obtaining models of Set Theory, using models of Determinacy. The primary application is the identification of a canonical model of Set Theory in which the Continuum Hypothesis is false. Such models have been sought for in the 35 years since Cohen's discovery of the method of forcing. The new model belongs to a large class of similarly obtained models. The basic machinery for the analysis of these models is developed in some detail through the study of the canonical model and several of the related models. A number of applications in combinatorial set theory are discussed. This is a research monograph the results being presented have not been published elsewhere. However the essential background material is also presented, making the account accessible to advanced graduate students in Mathematical Logic and Set Theory.