This present book provides the reader with a systematic exposition of the basic ideas and results of distribution theory and its applications to Fourier analysis and partial differential equations without using much sophisticated concepts of functional analysis. The treatment is properly motivated and simple but without any sacrifice to rigour. Examples are provided to illustrate the concepts; exercises of various level of difficulty are given at the end of each chapter. Unlike other elementary texts, the book covers important topics: basic properties of distributions, convolution, Fourier transforms, Sobolev spaces, weak solutions, distributions on locally convex spaces and on differentiable manifolds. It is a largely self-contained text. The text is based on graduate lectures given over a number of years. It can be used for senior graduate and graduate courses in mathematics and mathematical physics. This book would be an excellent choice for students, teachers and research workers in mathematics, physics and engineering, seeking a concise introduction to the subject and its important applications.