Digital holographic microscopy is a combination of digital holography and microscopy, combining the advantages of both technologies, which provides a non-destructive, label-free, accurate and near real-time measurement method for quantitative 3D measurement in the micro-nano field. However, phase aberrations are additionally introduced due to component defects, misalignment, and environmental disturbances in the digital holographic microscopy measurement results. To obtain accurate quantitative phase measurement results, the origin and principal components of aberrations must be analyzed, and then compensate and corrected for aberrations. This review began with an introduction to the main sources and effects of aberrations in digital holographic microscopy imaging measurement. Then, the existing aberration correction methods were reviewed based on their different workflows and characteristics. Finally, the future development direction of aberration correction was prospected, which would provide useful references for researchers engaged in digital holographic detection research.