This book is devoted to the field of fractional partial differential equations and their numerical solutions. It consists of five chapters, a preface, and a bibliography. Chapter 1 contains some physical background on the rich theory that was developed in recent years and it includes disciplines such as physics, biology and chemistry that are used in dynamical systems. The purpose of this chapter is to introduce the origin of the fractional derivative. Chapter 2 introduces definitions and basic properties of fractional derivatives, including the Riemann-Liouville fractional derivative, the Caputo fractional derivative, and the fractional Laplace operator. In Chapter 3, fractional partial differential equations are discussed. These include fractional diffusion equations, the Schrödinger equation, the fractional Ginzburg-Landau equation, the fractional Landau-Lifshitz equation, the fractional QG equation, the fractional Boussinesq approximation, and boundary value problems. The last three chapters deal with the numerical approximation of fractional calculus, algorithms for the Riemann-Liouville fractional derivative, numerical methods for fractional ordinary differential equations, and numerical methods for fractional partial differential equations.