Unlike conventional continuous-wave lasers with wide spectra, the amplification of single-frequency lasers in optical fibers is much more difficult owing to the ultra-high power spectral density induced nonlinear stimulated Brillouin scattering effect. Nevertheless, over the past two decades much effort has been devoted to improving the power scaling and performance of high-power single-frequency fiber amplifiers. These amplifiers are mostly driven by applications, such as high precision detection and metrology, and have benefited from the long coherence length, low noise, and excellent beam quality of this type of laser source. In this paper, we review the overall development of high-power single-frequency fiber amplifiers by focusing on its progress and challenges, specifically, the strategies for circumventing the stimulated Brillouin scattering and transverse mode instability effects that, at present, are the major limiting factors of the power scaling of the single-frequency fiber amplifiers. These factors are also thoroughly discussed in terms of free-space and all-fiber coupled architecture. In addition, we also examine the noise properties of single-frequency fiber amplifiers, along with corresponding noise reducing schemes. Finally, we briefly envision the future development of high-power single-frequency fiber amplifiers.