In this chapter we consider the approximation by the finite element method of variational equations and inequalities. In Chapter 6 we have reviewed some standard results for the well-posedness of variational equations and inequalities. The results can also be applied to the corresponding discrete problems over finite-dimensional spaces; in this way, we can then conclude the well-posedness of the discretized variational equations and inequalities. As we will see, C a's lemma (Theorem 10.1) reduces the task of estimating finite element solution errors for an elliptic variational equation problem to that of estimating approximation errors. For approximations of variational inequalities, we will show results of the type of C a's lemma. Then an application of the theory of finite element interpolation error estimates reviewed in Chapter 9 provides order error estimates for finite element solutions of variational equations and inequalities. Some references on finite element approximations of variational equations have been mentioned in Chapter 9. For detailed accounts on numerical solutions of variational inequalities, the reader may consult, among others, Glowinski, Lions, and Tr moli res ^[45], Glowinski ^[44], Kikuchi and Oden ^[70], Hlav ?ek, Haslinger, Ne?as, and Lov ek ^[61], and, more recently, Haslinger, Hlav ?ek, and Ne?as ^[57].