The article investigates the present review of thermoelastic theories of circular and rectangular objects with moving heat sources published on paper after 1950. The main theme of thermoelastic problems on circular and rectangular structures is to establish operational methods to solve governing differential equations. In the thermoelastic problems, we have considered a few practical problems of technical interest, taking into account the circular and rectangular shape geometry with moving heat sources by using integral transform techniques. AMS Subject classifications: 35B07; 35G30; 35K05: 44A10 Keywords: circular and rectangular objects, temperature distribution, thermal stresses, integral transform.