When the publisher Chapman & Hall invited us to write a book for its series in Applied Mathematics and Mathematical Computation, the best we could contribute appeared to be an outline of the work done in the last two decades on the nonlinear thermoelastic theories for twinning and phase transitions in crystalline materials. The developments of this elastic approach to crystal mechanics, rich in new theoretical aspects as well as applications, have been so extensive and ramified that we felt a smooth, wellhoned presentation of them might be difficult to obtain. However, the very abundance of and interest in the results already available were well worthy of an organized survey covering the essential notions on the subject. Our effort has been to fulfill in this way some of the needs of the ever-increasing, strongly interdisciplinary community devoted to this highly interesting research area, which stretches from deep theoretical issues in mathematics to very practical problems in materials science. We briefly touched upon various subjects, such as crystallography, bifurcation theory, Landau theory, variational calculus, which are natural tools for the development of the model. Of course, we made no attempt to be comprehensive, also because a great deal of investigation is currently being carried out in many different directions to further clarify and broaden the scope of the proposed theory.