As we discussed in chapter 1, one of the main ways in which crystalline materials deform under an external action or along a displacive phase transformation is by mechanical twinning, to which we will mostly refer henceforth when using the word twinning.1 In this chapter we assume crystals to be stress free, under neither loads nor displacement boundary conditions, and define mechanical twinning deformations as special pairwise homogeneous natural states (that is, stable, stress-free equilibria) for a crystal involving any pair of symmetry-related minimizers of its free energy (see Ericksen (1977), (1981a), (1987), Parry (1980), James (1981), (1984a), Gurtin (1983), Pitteri (1985b); also Ball and James (1987) and Bhattacharya (1991)). Within the elastic model considered in chapter 6, twins are the simplest nonhomogeneous minimizers of the functional (6.12), for no load or boundary displacement imposed. As we will see below and in the next chapter, when the energy density has the invariance properties described in §6.3.1 nonlinear elasticity becomes flexible enough to describe twins as well as more complex formations, to be called microstructures (chapter 10), which are produced to accommodate external actions.