Maxwell's Equations are one of the most important models in different fields.They describe electromagnetic phenomena such as micro,radio and radar waves.The modeling of systems involving electromagnetic waves is widely spread.In this book I present some important analytical and numerical aspects of Maxwell's Equations.I put a review of some basic properties of electromagnetic theory such as: electromagnetic field, electromagnetic static field time varying fields.I use some well-known analytical methods in solving these equations in some particular domain such as a sphere and circular cylinder.These analytical methods include Separation of variables,Series expansion method,conformal mapping and integral methods such as Laplace transforms and sine and cosine transforms.Numerical methods for solving Maxwell's equations are widely used nowadays and are referred to Computational Electro-Magnetic (CEM) here I use the Finite Difference and Finite Difference Time Domain Method (FDTDM). And I also proposed the Yee Algorithm, and the convergence and stability for these numerical methods are investigated.