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Compressor is the crucial device for the present fuel cell system, reflected by the compressor's selection, performance, cost, and control. The review papers on compressor usually focus on its category and the pros and cons in commercial use. This paper reexamines the necessity of the air compression of fuel cell system and systematically reviews the compressor progress and application from criteria for air compression to compressor control. The leading six parts of this paper answer, respectively, that which kind of stack is suitable for compression, which compressors are commonly used in China market, what features should the compressor have for the fuel cell application, which phase is the current compressor in and will be in, how to effectively control the compressor. Of them, the analyzing model for the criteria to conduct compression is firstly proposed because it is a generalized method and independent of experimental data. Using the thermal dynamics to analyze the phase and possible ceiling of the compressor is another novelty (DOE or any other organizations will not give this ceiling and analysis). Finally, the compressor control at the angle of system efficiency improvement is firstly reviewed, as well. This paper is beneficial for the researchers to know the overall state of the compressor and is of guidance for them to design a more efficient and commercial fuel cell system. • Generalized data-free analysis model for criteria to compress air is firstly given. • Analyzing the phase and development ceiling of the compressor. • Comparison of technique similarity between fuel cell and inner combustion engine. • Compressor control at the angle of system efficiency promotion is firstly reviewed. • Sliding mode control and extremum seeking control are promising for compressors. [ABSTRACT FROM AUTHOR]