Additive manufacturing (AM) is characterised by several unique advantages, such as (freedom of) design, capability of fusing dissimilar materials, near-net-shape, and achieving a more sustainable production. While the increased precision of metal AM in recent years reduced the needed amount of post-processing to meet dimensional tolerance, the requirements for functional surfaces necessitate a well-understood post-processing, ranging from heat treatment to machining and finishing. The inherently rough initial (as-built) surface topography next to complex material microstructure affects the capability of post-processing/finishing operations to smooth the surface texture and obtain a favourable surface integrity. In this respect, a more fundamental understanding of the effects of material properties on post-processing/finishing is needed. Therefore, this review paper aims to establish the relationship between the characteristics of different AM technologies, microstructural properties of materials in as-built and heat-treated conditions, and the physical properties influencing the response of additively manufactured materials during post-processing/finishing operations. In particular, emphasis is placed on the physics-based understanding of how the microstructural characteristics of 316L, Ti6Al4V and Alloy 718 produced using the two principal technologies, Powder Bed Fusion (PBF) and Direct Energy Deposition (DED), influence their mechanical properties like tensile strengths, hardness and ductility. These properties are among the key factors influencing the response of material during post-processing/finishing operations involving material removal by shear deformation. This review paper also discusses the role of post-processing/finishing on fatigue performance, tribological behaviour and corrosion resistance of investigated AM materials. The paper summarises the state-of the art of post-processing/finishing operations and future research trends are highlighted.