The book is a carefully written handbook of real analysis, which will be very helpful for engineers, physicists and anybody who wants to use mathematics in applications. It contains descriptions of many important results with illustrative examples. Starting with the basics of real analysis, it explains topology, continuity, integration, up to uniform convergence, the Weierstrass theorem and similar topics. Special attention is paid to applications in differential equations and in Fourier analysis. On the other hand, deeper parts of theory are omitted. The last 25 pages contain a glossary of terms from real analysis, a detailed list of notations, a guide to literature, index and bibliography. For a student of mathematics, it could serve as a survey of results that should be mastered during undergraduate studies.