This book offers a clear presentation of major topics in real and complex analysis. Intended for undergraduate mathematics and engineering students, it covers the essential analysis that is needed for the study of functional analysis, developing the concepts rigorously with sufficient detail and with minimum prior knowledge of the fundamentals of advanced calculus required. Divided into seven chapters, it discusses the proof of the prime number theorem, Picard's little theorem, Riemann's zeta function, Euler's gamma function and Riemann hypothesis. It also addresses the applications of complex analysis to analytic number theory in a manner that does not require any prior exposure to number theory. Further, it includes extensive exercises and their solutions with each concept. The book examines several useful theorems in the realm of real and complex analysis, most of which are the work great mathematicians of the 19th and 20th centuries, such as Baire, Banach, Cauchy, Dirichlet, Fatou, Fourier, Fubini, Hadamard, Jordan, Lebesgue, Liouville, Minkowski, Morera, Picard, Poisson, Radon, Riemann, Riesz, Schwarz, Taylor, Weierstrass, and Young, to name just a few.