The maximum entropy production principle (MEPP) was repeatedly and independently proposed in the mid-twentieth century in various fields of physics and proved to be extremely effective in considering various non-equilibrium problems. The review describes the main areas of investigations that formed the basis of this principle, provides the modern formulation of MEPP and its restrictions. Much attention is paid to the consideration of non-equilibrium phase transitions based on MEPP, as well as the relationship of the principle with other known statements about the behavior of entropy. The role of the principle in the consideration of various modern problems in both physics and biology, in particular, carcinogenesis, the laws of evolution and the definition of life, is discussed.