This is a graduate course for students who are interested in the theory of partial differential equations. The main goal of the course is to familiarize students with basic analytic techniques in studying partial differential equations. Unlike the PDE course for undergraduates, we will focus more on the fundamental questions such as existence, uniqueness, regularity, etc. We shall begin with harmonic functions and the Laplace equation. The study of second order elliptic equations is the main theme of the course. In addition, we will also study hyperbolic and parabolic equations.