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 Chapter the thermal effects due to motion are investigated. Temperature is introduced from the outset. The system entropy balance, equivalent to the energy balance, is a variational relationship which is linear with respect to the actual temperature. It is extended by induction to virtual temperatures giving the principle of virtual thermal power. The usual theory requires the first gradient of the temperature but experiments, for instance the engineering thermal insulation of houses, show that a second gradient theory is needed. The evolution of the temperature is due to external entropy source and to an internal entropy source: the dissipation. The dissipation partly due to mechanical effects couples the thermal and mechanical evolutions. There are internal constraints: physical relationships between state quantities. A systematic way to detect them is given. Among them there are the compatibility conditions which are not always satisfied due to defects and dislocations. In this case the polar decomposition introduces the curl of a matrix accounting for the defects. A condition for its validity will be given by a constitutive law based on experiment: when the actions are not too large, the compatibility conditions are satisfied and when the actions are large the compatibility conditions are not satisfied.