Selective laser melting (SLM) is a powder bed fusion additive manufacturing technology that allows the production of high-performance metal parts with geometrically complex shapes, such as curved surface, can be used in practical engineering applications. In recent years, systematic attempts have been made to uncover the material-process-microstructure-property linkage in SLM-fabricated planar-surface metal parts; however, little is known on the performance and quality of SLM-fabricated metal parts with curved surfaces. In this paper, we studied the effects of build orientation on the geometric performance of SLM-built 316L stainless steel (SS) samples with a sinusoidal curved surface. The results indicated that the average values of profile roughness Ra and surface texture attributes Sa, Sk, and Vmc decreased with the increase in build orientation. Moreover, the locations at curvatures C2 and C4 exhibited the best surface finish at the build orientation of 75 deg and 90 deg. In addition, the least square method was used to fit the actual profile of the curved surface of as-built samples fabricated at each build orientation, and it was found the actual profile demonstrated the best consistency with the designed one at 90 deg build orientation. In this paper, the process–curvature–geometric performance relationship of SLM-built metal parts with curved surface was uncovered. In addition, this paper provides keen insights into assessing the geometric performance of SLM-built curved-surface metal parts and establishes a roadmap toward SLM-fabrication of metal parts with complex geometries for practical engineering applications.