Heat is an abundant but often wasted source of energy. Thus, harvesting just a portion of this tremendous amount of energy holds significant promise for a more sustainable society. While traditional solid-state inorganic semiconductors have dominated the research stage on thermal-to-electrical energy conversion, carbon-based semiconductors have recently attracted a great deal of attention as potential thermoelectric materials for low-temperature energy harvesting, primarily driven by the high abundance of their atomic elements, ease of processing/manufacturing, and intrinsically low thermal conductivity. This quest for new materials has resulted in the discovery of several new kinds of thermoelectric materials and concepts capable of converting a heat flux into an electrical current by means of various types of particles transporting the electric charge: (i) electrons, (ii) ions, and (iii) redox molecules. This has contributed to expanding the applications envisaged for thermoelectric materials far beyond simple conversion of heat into electricity. This is the motivation behind this review. This work is divided in three sections. In the first section, we present the basic principle of the thermoelectric effects when the particles transporting the electric charge are electrons, ions, and redox molecules and describe the conceptual differences between the three thermodiffusion phenomena. In the second section, we review the efforts made on developing devices exploiting these three effects and give a thorough understanding of what limits their performance. In the third section, we review the state-of-the-art thermoelectric materials investigated so far and provide a comprehensive understanding of what limits charge and energy transport in each of these classes of materials.