This solutions manual is geared toward instructors for use as a companion volume to the book, A Modern Theory of Integration (AMS Graduate Studies in Mathematics series, Volume 32). A comprehensive, beautifully written exposition of the Henstock-Kurzweil (gauge, Riemann complete) integral. The theory of integration is one of the twin pillars on which analysis is built. The first version of integration that students see is the Riemann integral. Later, graduate students learn that the Lebesgue integral is better because it removes some restrictions on the integrands and the domains over which we integrate. However, there are still drawbacks to Lebesgue integration, for instance, dealing with the Fundamental Theorem of Calculus, or with improper integrals .