The entransy and entransy dissipation extremum principle proposed have opened up a new direction for the heat transfer optimization. The emergence and development of entransy theory are reviewed. Entransy theory and its applications are summarized from several aspects, such as heat conduction, heat convective, heat radiation, heat exchanger design and mass transfer, etc. The emphases are focused on four aspects, i.e., the comparison between entropy generation rate and entransy dissipation rate, the combination of entransy dissipation extreme principle with finite time thermodynamics, the combination of entransy dissipation extreme principle with the heat conduction constructal theory, and the combination of entransy dissipation extreme principle with the heat convective constructal theory. The scientific features of entransy theory are emphasized. Influenced by oil crisis in the 1970's directly, the heat transfer enhancement was world-widely paid attention by the scientific community, and was quickly developed into one of very important branches in thermal science and technology fields. With the progress of social development ideas, such as sustainable development, low-carbon technology and low-carbon economy, the concept of heat transfer enhancement has developed into a broader scope concept of heat transfer optimization [1]. In the past 30 years, the basic theory of heat transfer optimization has been made considerable progresses, and finite-time thermodynamics, field synergy principle, constructal theory as well as entransy theory are the representative achievements. The emergences of these theories promoted the developments of thermodynamics and heat transfer. Finite-time thermodynamics, which combined thermodynamics, heat transfer and fluid mechanics, provided a theoretical fundament for performance optimizations of practical processes and devices with the constraints of a finite size and finite time. Field synergy principle unified the essential physical understandings of convective heat transfer and heat transfer enhancement phenomena, and provided a theoretical fundament for the development of heat transfer enhancement technology to achieve better energy saving effects. Constructal theory, which was called the thermodynamics of non-equilibrium system with configuration in the field of thermal science, provided a theoretical fundament for the unified descriptions of the generation of flow structures in natural organization and the designs of various flow structures. Fourier's law of heat conduction, Newton's law of cooling and Stephen-Boltzmann law of blackbody radiation only gave the concept of heat transfer rate, while the concepts of entransy and entransy dissipation lay the foundation of defining the concept of heat transfer efficiency which did not appear in the heat transfer theory before. Entransy theory provided a new theoretical fundament for heat transfer optimization, which is different from that of the entropy generation minimization.