Every day we experiment motion and shape change of continuum media, assembly of different space dimension structures, fluids, solids which exhibit numerous evolutions either particular or general: large deformations, incompressibility, cavitation, contact, surface tension, defects, dislocations,... Stretch and rotation are a common feature of these motions and phenomenons. We base a continuum mechanics predictive theory on stretch and rotation quantified by matrices. The mechanical and thermal evolutions of a solid are investigated in this Chapter. The solid may be incompressible: a compression cannot modify its volume but cavitation may occur as experimented in the water hammers. A constitutive law relates an internal stress to the matrix accounting for the defects and the dislocations. It quantifies the conditions for defects and dislocations to appear. It is proved that in a rigid body motion the defect stress does not work. Thus the power of the internal forces remains null in a rigid body motion. Extension to damage, shape memory alloys and Cosserat materials are described. A Cosserat material depends on an extra rotation matrix. With the internal constraint that this extra rotation matrix is equal to the polar decomposition rotation matrix, the Cosserat material theory upgraded to a third gradient theory coincides with the third gradient theory developed in this book.