We analyze the mathematical problems that arise during identification of structured and viscoelasic media on the basis of nonlinear integral models. We consider a variety of models: the nonlinear integral model of the K-BKZ type, the integral and differential Maxwell models, the Jeffreys integral model and its nonlinear generalizations, namely, the Oldroyd integral models of the A and B types. We introduce the nonlinear integral model by Slemrod that employs the direct integral operator of the Gammershtein type. We make a conclusion that the problem of identification (that is, the problem of construction of the state equations for a medium) may be solved on the basis of all the models mentioned when using the hybrid theoretical and experimental approach. The experimental data may be obtained by using rheological apparatus. The problems of identification are often of ill-posed type by Hadamard and consequently special methods of regularization should be developed for solving such problems. The results of computational experiments are compared with the results of experimental tests and the results of full-scale tests for viscoelastic and electorheological media. We propose an adaptive hierarchic model approach for describing both linear and nonlinear characteristics of systems under consideration.