Besides its advantages in volume, power and beam quality, a monolithic integration Master-Oscillation Power-amplifier (MOPA) can also realize a narrower linewidth and dynamic single-mode by integrating Bragg grating. Its application value is high in the fields of frequency doubling, pumping, optical communication and sensing, which makes it a popular research topic in recent years. This paper firstly went over the mainstream structure and characteristics of monolithic integrated MOPA, including a tapered amplifier, ridge amplifier, Bragg grating and three-section MOPA. Based on their working principles and performance characteristics, we introduce the main research directions and the latest development trends in combination with their problems. Aiming at the problem of beam quality degradation at high power in monolithic integrated MOPA, the optimal design of epitaxial layer structure, facet optical film and electrode aspects are then summarized for monolithic integrated MOPAs. After that, we sort out the research progress of MOPAs with different performance characteristics for various application requirements including high power, narrow linewidth, high beam quality and high brightness. Finally, we prospect the development trend of monolithic integrated MOPA.