In this chapter we examine the concept of constitutive equation gap (CEG) as a tool for the identification of parameters associated with behavior models for solid materials. The concept of CEG was initially proposed for error estimation in the finite element method. It then turned out to be also a powerful tool for identification, especially with many applications in model updating. Essentially equivalent concepts have been proposed in other contexts for solving inversion problems, such as the elec- trostatic energy functionals of Kohn and Vogelius. Two important characteristics of CEG functionals are (i) their strong and clear physical meaning, and (ii) their additive character with respect to the structure, allowing the definition of local error indicators over substructures.