The entropy generation minimization is widely used to deal with optimization problems of heat transfer and heat-work conversion. However, it is found that the minimization of entropy generation does not always lead to the optimization of the design objectives in engineering. So, it is necessary to discuss the optimization direction and application preconditions of the entropy generation minimization. In this paper, we study this topic both theoretically and numerically. Our analyses show that the concept of entropy generation directly measures the exergy loss or the ability loss of doing work, so the optimization objective of the entropy generation minimization is to minimize the exergy loss and maximize the ability to do work for the optimized system. However, we have different design objectives in engineering, such as the maximum heat transfer rate, the maximum heat exchanger effectiveness, the minimum average temperature of the heated domain, the maximum output power, the maximum coefficient of performance of heat pump systems, the homogenization of temperature field, etc. Not all of these objectives are consistent with the optimization direction of the entropy generation minimization. Therefore, it is reasonable that the entropy generation minimization is not always applicable. Furthermore, when the relationship between entropy generation and design objective can be set up, the application preconditions of the entropy generation minimization are also discussed. When the preconditions are not satisfied, the entropy generation minimization does not always lead to the best system performance, either. Some examples are also presented to verify the theoretical analyses above. For heat transfer, a one-dimensional heat transfer problem and the entropy generation paradox in heat exchanger are analyzed. For the one-dimensional heat transfer problem, the entropy generation minimization leads to the minimum heat transfer rate when the temperature difference between the boundaries is fixed. Therefore, if our design objective is the maximum heat transfer rate, the entropy generation minimization is not applicable. When the heat transfer rate is fixed, smaller entropy generation rate leads to higher boundary temperature. Therefore, if our design objective is to reduce the boundary temperature, the entropy generation minimization is not applicable, either. For the entropy generation paradox, it is shown that the concept of entropy generation cannot describe the heat transfer performance of heat exchangers. Therefore, the paradox still exists and has not been removed to date. This is verified by the theoretical analyses and the numerical simulation for a parallel flow heat exchanger in which the irreversibility from the pressure drop can be ignored. For heat-work conversion, the energy flow and the exergy flow are analyzed. According to the analyses, we discuss the applicability of the entropy generation minimization to the heat-work conversion system in which the output power, the heat-work conversion efficiency and the thermo-economic performance are taken as the optimization objectives. It is also shown that the application of the entropy generation minimization is conditional. In a word, the discussion on the examples verifies the theoretical analyses.