This chapter is devoted to the study of finite element methods in one and two dimensions. We begin by presenting the general theory of Galerkin methods and their analysis; in particular Galerkin orthogonality and Cea’s lemma are introduced in an abstract setting. Then the construction of finite element spaces, and their bases, in one dimension is detailed. The notions of mesh and hat basis functions are introduced here. The general theory of Galerkin approximations is then used to reduced the error analysis of finite element schemes to a question in approximation theory. The properties of the Lagrange interpolant in Sobolev spaces (in one dimension) then close the argument. Duality techniques, i.e. Nitsche’s trick are then used to obtain optimal error estimates in L2. The same ideas are presented, mostly without proof, for the finite element scheme in two dimensions.