Our goal in this chapter is to analyze the well-posedness of finite element methods. Of course, the sensitivity of numerical methods to perturbations in their data is governed partly by the sensitivity of the underlying equations being solved. Thus we will need to study the existence, uniqueness and continuous dependence on the data for elliptic boundary value problems. These issues will be described in terms of some basic tools of functional analysis, beginning with function norms and spaces in Sections 5.1.1 and 5.1.2. Appropriate function spaces will allow us to describe function derivatives in Section 5.1.3, and function norms that involve derivatives in Section 5.2. These derivative norms allow us to bound the solution of elliptic partial differential equations in terms of features of the problem domain, coefficients in the differential equation and boundary values. Once we have described elliptic equations in Section 5.3 and developed some theory of well-posedness for elliptic boundary value problems in Section 5.4, we will be able to discuss some basic ideas regarding the convergence of finite element methods in Section 5.5. However, some important details of this finite element convergence theory will be delayed until Chapter 6. Those details concern polynomial approximation, coordinate maps, domain approximation, quadrature rules, and numerical treatment of boundary conditions.