These two papers resume the theoretical background supporting the main ideas of the exergy cost accounting and the thermoeconomic approach followed by Valero and co-workers. Part I introduces the basic requirements. with a simple example accompanying the dissertations, to calculate the exergy and thermoeconomic costs and to perform the thermoeconomic analysis of a complex system. The connections with other thermoeconomic approaches and schools are briefly explained. Part II presents, as an illustration of the applications of thermoeconomic analysis, some of the most interesting applications of costs to the operation diagnosis and optimization of a complex system, showing the results on the mentioned example presented in Part I.