This short course introduces the fundamental concepts of Electromagnetic theory as embodied in Maxwell's equations. Following a short discussion of Maxwell's equations in free space, and the definition of the time harmonic Maxwell Equations, we will discuss the classical boundary value problems, which arise in scattering theory. After defining these problems and establishing the abstract uniqueness of solutions, we will describe various methods for representing solutions to the time harmonic equations using layer potentials, leading to the so-called, Boundary Integral Equation Method (BIEM). Following a short discussion of classical Fredholm theory, we show how these representations lead to a variety of numerical methods for the approximate solution of scattering problems. The course concludes with a short introduction to the Fast Multipole Method (FMM) of Rokhlin and Greengard, which has made it possible to solve large, interesting problems using the BIEM.