Breathers, nonlinear waves with periodic evolution or periodic distribution, whose shapes undergo periodic oscillations during propagation, can be localized in time or space with a finite background. Such breathers are widely found in physical systems such as nonlinear optics, fluid mechanics, Bose-Einstein condensates and PT-symmetric systems. Usually, according to the different excitation background, it can be divided into zero background and plane wave background. Among them, breathers on zero background are essentially “multi-soliton bound states”, and there are extensive experimental reports in ultrafast fiber lasers. Here we emphasize the importance of the breathers on the plane wave background of the latter case. Indeed, these breathers exhibit periodic energy exchanges with the plane wave background.Common breathers are mainly classified by their properties into Kuznetsov-Ma breather, Akhmediev breather, Tajiri-Watanabe breather, and Super-regular breather. Quite remarkably, the Akhmediev breather can be used to describe the well-known Fermi-Pasta-Ulam recurrence and higher-order modulation instability, showing the dynamic processes with doubly periodic characteristics such as regular or shifted recurrence and pulse splitting. In addition, the study of breathers is helpful to understand the phenomena including the generation of supercontinuum and turbulence in nonlinear systems. Moreover, the interactions of the breather have been shown to be useful for high-power pulse preparation and to reveal the nonlinear evolution of modulation instability.Generally, the study of the breather dynamics is mainly based on the integrable (1+1) dimensional nonlinear Schr?dinger equation. In brief, the existence and excitation conditions of different kinds of breathers are obtained theoretically by exact solution in the integrable system, and the different breather structures are obtained by using the simple initial states for evolution. On the other hand, the formation conditions of breathers can be analyzed through the excitation mechanism of different breathers, and the dynamic properties and experimental realization of breathers can be explored. At present, the theoretical and experimental results of breathers are mainly discussed from the following aspects: 1) the existence conditions of breathers in different nonlinear physical systems; 2) theoretical method for accurate solution of breathers; 3) structure and properties of breathers; 4) excitation mechanism and production conditions of breathers; 5) evolution characteristics of different breathers; 6) interaction characteristics of breathers; 7) applications of breathers in supercontinuum generation, high-power pulse preparation, etc.For nonlinear physical system, nonlinear fiber is a mature nonlinear experimental platform in experimental science. Therefore, based on nonlinear optical systems, this paper briefly discuss the experimental and theoretical research progress of several typical breathers on continue wave background. Based on the analytical expressions of fundamental breathers, the dynamical properties of Kuznetsov-Ma breather, Akhmediev breather, and Peregrine rogue waves in time and frequency domains are enumerated in detail. The historical development and practical physical significance of breathers is also discussed. The excitation mechanism and generation conditions of the fundamental breathers are discussed based on the modulation instability in nonlinear systems. The results show that Kuznetsov-Ma breather and Akhmediev breather can be excited by localized perturbation with zero modulation frequency and periodic perturbation with non-zero modulation frequency, respectively. Furthermore, the quantitative conversion relationship between different breathers is analyzed by combining effective energy and modulation instability. Finally, we show the dynamics of breathers interaction on a plane wave background. The phase sensitive collision of two breather is reported to form three different localized structures, that is super-high peak rogue waves, quasi-annihilation super-regular breathers, and ghost structures without any change. These important breather structures have been confirmed and discussed in detail in theoretical and optical experiments. Actually, understanding the physical significance of fundamental breathers and their interactions is expected to play an important role in the study of rogue wave events generation, integrable turbulence, modulation instability, Fermi-Pasta-Ulam recurrence and other phenomena in nonlinear systems.