The current state-of-the-art metal additive manufacturing (AM) process still cannot meet the high industry requirements in terms of surface roughness. In addition, there are limited ISO/ASTM standards on the post-processing of metal AM components. Considerable efforts have been made to close these current gaps and challenges for standardization. In this review, First, all the existing ISO/ASTM standards under the jurisdiction of ASTM Committee F42 on additive manufacturing technologies are briefly discussed. Second, two of the most promising additive manufacturing processes: powder bed fusion (PBF) and directed energy deposition (DED), as well as their common defects of AM components, are thoroughly reviewed. The build quality, surface finish characteristics, and dimensional integrity of the as-built AM components are still not satisfactory (i.e., Ra > 5 µm and Rz > 100 µm). Third, the review is focused on identifying potential solutions to surface-finish complex additive manufactured geometries and reduce the surface roughness to meet the industry requirements (i.e., Ra < 1 µm and Rz < 20 µm). Current state-of-the-art surface finishing techniques, their material removal mechanisms, process parameters, advantages, and limitations to post-process PBF components are discussed. The technical gaps and future perspectives in enhancing the surface quality of AM components are outlined.