This is a one year course that is intended to be a graduate level introduction to the theory of partial diferential equations (PDEs). The course material will start with an overview of the basic properties of the wave equation, Laplace's equation and the heat equation; introducing Fourier transform techniques, distributions, Green's functions, and some of the basic notions of the theory of PDE. We will proceed to cover the general theory of PDE, including first order theory, the Cauchy Kowalevskaya theorem and generalizations, the Malgrange - Ehrenpreis theorem, and subsequent counterexamples to existence. On a more general level, we will take up the theory of elliptic equations and their regularity, symmetric hyperbolic systems and energy estimates, and parabolic systems. We will then move to the study of more advanced techniques, such as the development of Brownian motion and Wiener measure, pseudodifferential and Fourier integral operators, and methods of nonlinear functional analysis. Throughout this course, an attempt will be made to connect the theory to relevant examples of current research interest in mathematics and its physical applications.