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This first book of two volumes reviews the basic models and theories of nuclear structure and gives an in-depth analysis of their experimental and mathematical foundations. It shows the relationships between the models and exhibits the value of following the strategy of: looking for patterns in all the data available, developing phenomenological models to explain them, and finally giving the models a foundation in a fundamental microscopic theory of interacting neutrons and protons. The book provides a unique resource and an exposition of the essential principles, mathematical structures, assumptions, and observational data on which the models and theories are based. It avoids discussion of many non-essential variations and technical details of the models. "Volume II: Unified Models" shows the relationship between the macroscopic and microscopic perspectives of nuclear structure theory and presents the essentials of modern developments.