The Calculus of Variations is one of the classical branches of mathematics, which concerns with maximizing or minimizing a functional. We will start with the isoperimetric problem, which is one of the oldest problems in mathematics. Then we will formulate a general minimization problem and a direct method to solve the problem. After introducing the Sobolev Space and some Functional Analysis, we will prove the existence and uniqueness of the Dirichlet problem. If time permits, we will discuss the regularity of the solution.