Nearly every Ph.D. student in mathematics needs to take a preliminary or qualifying examination in real analysis. This book provides the necessary tools to pass such an examination. Clarity: Every effort was made to present the material in as clear a fashion as possible. Lots of exercises: Over 400 exercises, ranging from routine to challenging, are presented. Many are taken from preliminary examinations given at major universities. Topics covered include the definition and construction of measures, the Lebesgue integral and its properties, limit theorems, product measures, signed measures, the Radon-Nikoydym theorem, differentiation, LP spaces, Fourier transforms, the Riesz representation theorem, and Banach and Hilbert spaces. Additional topics include topology, probability, harmonic functions, Sobolev spaces, singular integrals, spectral theory, and distributions