REAL ANALYSIS: An Introduction to the Theory of Real Functions and Integrationis intended to be used in a two-semester course on Real Analysis/Real Variables at the graduate level. It is self-contained, with coverage included on topology, measure theory and integration. It includes many applications from the fields of science and engineering, thus serving a broad audience. The topics addressed include Ascoli's theorem, Stone-Weierstrass approximation, and applications of Fubini's theorem.