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Geared toward upper-level undergraduates and graduate students, this treatment of geometric integration theory consists of three parts; an introduction to classical theory, a postulational approach to general theory, and a final section on Lebesgue theory. The opening third of the treatment leads to the theory of the Riemann integral and includes a study of smooth manifolds. The second part explores abstract integration theory, some relations between chains and functions, general properties of chains and cochains, and chains and cochains in open sets. The final section surveys Lebesgue theory in terms of flat cochains and differential forms, Lipschitz mappings, and chains and additive set functions. Useful appendixes and indexes conclude the text.