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. In this introduction we sum up the elements and their articulations to achieve a framework to develop predictive theories of motion and of thermal evolution of a continuum medium. There are few formulas but we focus on the reasons of our choices and partis pris supported by experiment. Among them, let us mention the choice to describe the motion with the stretch and rotation matrices of the classical polar decomposition; the choice of the third gradient theory for the power of the internal forces and for the power of the acceleration forces; the choice of the second gradient theory for the temperature in the energy balance equation. The computation of the position depending on the stretch and rotation matrices shows that defects and dislocations can appear. They are absent if an internal stress is not too large. In the different Chapters we present theories which apply to many continuum media either solid or fluid.