The historical background, research development, and the state-of-the-art of finite time thermodynamic theory and applications are reviewed from the point of view of both physics and engineering. The emphasis is on the performance optimization of thermodynamic processes and devices with finite-time and/or finite-size constraints, including heat engines, refrigerators, heat pumps, chemical reactions and some other processes, with respect to the following aspects: the study of Newton's law systems, an analysis of the effect of heat resistance and other irreversible loss models on the performance, an analysis of the effect of heat reservoir models on the performance, as well as the application for real thermodynamic processes and devices. It is pointed out that the generalized thermodynamic optimization theory is the development direction of finite thermodynamics in the future.