The design of heat exchangers may change dramatically through the use of additive manufacturing (AM). Additive manufacturing, colloquially known as 3D printing, enables the production of monolithic metal bodies, devoid of contact resistance. The small volume of the exchanger, its lightness of weight, and the reduction of its production costs, compared to conventional methods, make the production of heat exchangers by AM methods conventional technologies. The review study presents a new look at the TPMS as a promising type of developed surface that can be used in the area of heat transfer. (Thus far, the only attractive option. The most important feature of additive manufacturing is the ability to print the geometry of theoretically any topography. Such a topography can be a minimal surface or its extended version—triply periodic minimal surface (TPMS). It was practically impossible to manufacture a TPMS-based heat exchanger with the method of producing a TPMS.) The issues related to the methods of additive manufacturing of metal products and the cycle of object preparation for printing were discussed, and the available publications presenting the results of CFD simulations and experimental tests of heat exchangers containing a TPMS in their construction were widely discussed. It has been noticed that the study of thermal-flow heat transfer with the use of TPMSs is a new area of research, and the number of publications in this field is very limited. The few data (mainly CFD simulations) show that the use of TPMSs causes, on the one hand, a several-fold increase in the number of Nu, and on the other hand, an increase in flow resistance. The use of TPMSs in heat exchangers can reduce their size by 60%. It is concluded that research should be carried out in order to optimize the size of the TPMS structure and its porosity so that the gains from the improved heat transfer compensate for the energy expenditure on the transport of the working fluid. It has been noticed that among the numerous types of TPMSs available for the construction of heat exchangers, practically, four types have been used thus far: primitive, gyroid, I-WP, and diamond. At the moment, the diamond structure seems to be the most promising in terms of its use in the construction of heat exchangers and heat sinks. It is required to conduct experimental research to verify the results of the CFD simulation.