In this paper we analyze relaxation phenomena of amorphous materials near or below the glass transition. A phenomenological theory is suggested that maintains the main ingredients of the widely accepted models, i.e., those quasi-universal properties of structural relaxation which are well established, while a new approach is adopted for constructing the overall relaxation under a given temperature history. The evolution of the relaxational part of a property p under time-temperature changes is described by a first order relaxational equation that states that the instantaneous advance of the relaxation is proportional to the amount of deviation from equilibrium. The model consistently combines three different principles: (a) Linearity of response, (b) time-temperature re-scaling, and (c) power law relaxation at short times. This is achieved by imposing the following requirements on the relaxational equation: that the equation be expressed as a unique function of the reduced time; and that it provides the Kohlrausch-Williams-Watts relaxation law in the particular case of a temperature jump experiment. In addition, the relaxation time is not a function of fictive temperature. This approach provides as an outcome a new type of superposition over past perturbations. The analysis of rate heating/cooling experiments shows that the model reproduces the hysteresis of the fictive temperature and the peaks in heat capacity curves frequently observed in experiment. The physical meaning of the shift relationship between cooling rate and fictive temperature is critically examined on the basis of scaling properties and relaxational properties and some limitations of the standard result are identified. A more general and physically reasonable relationship is obtained by rigorous derivation in the framework of the new model. It is therefore demonstrated that that relationship is not related to nonlinearity, contrary to what is widely believed. In addition, it is shown that the more general relationship involves the parameter beta describing the slowing down of the relaxation. This provides the basis for new relations to be inferred between apparently different phenomenological properties. An explanation is advanced for the observed correlations between measured parameters in the Tool-Narayanaswamy-Moynihan phenomenology.