The Finite Element Method(FEM) is one of many numerical methods to approximate solutions to ordinary and partial differential equations. FEM has been applied to numerous problems found in the fields of Fluid Mechanics, Electromagnetics,  Lagrangian Mechanics, etc. While there are many approaches to the Finite Element Method, I will present the Galerkin approach. I will discuss both theory and implementation of FEM, including the weak formulation and the idea of a weak versus strong derivative. I will show the FEM for the Sturm–Liouville ODE, the 2-D Poisson Equation. If time permits I will present a semi-discrete FEM for the 2-D Wave Equation. These problems will be solved using the Python library FENICS. I will also discuss the advantages and disadvantages of the method in practice.