"If the world is meaningless that prevents invent any sense?"-Lewis Carroll “Alice''s Adventures in Wonderland”. About 50 years ago importance to apply differential geometry for extending the elastic continuous medium model was recognized by researchers. The introduced affine-metric objects characterize the internal geometric structure of the continuous media and its difference from the Euclidean geometry. The affine- metric objects are the internal variables, and they can''t be measured directly. This book shows how to establish the relation between the non-Euclidean geometric parameters of description and experimentally measured characteristics. It is demonstrated on the basis of the non-equilibrium thermodynamics formalism that to determine the affine-metric characteristics it is experimentally sufficient to measure two independent functions: internal energy and dissipation function. The proposed approach allows us to construct a thermomechanical model of a continuous medium including a full set of non-Euclidean characteristics which corresponds, from the physical point of view, to the description of dislocations, disclinations and point defects.