Laser micromachining is widely used to fabricate planar functional surface microtextures, for its advantages of high quality, low cost and high efficiency. However, the application of laser micromachining in fabricating curved surface microtextures is hindered by the limited ablation ranges and frequent changes of beam incident angles, which result into low machining quality. In order to realize the laser micromachining of high-quality microtextures on complex curved surface, a multi-axis linkage laser micro-milling-based surface texturing method is proposed, which couples a multi-axis linkage precision motion platform with a nanosecond pulse laser beam. Furthermore, the strategy of constant coincidence between the laser beam and the surface normal is adopted to realize the high-quality laser micromachining of high precision microtextures on curved surface. Firstly, the projection law of planar surface microtexture on curved surface, as well as the kinematics of the position and attitude adjustment of laser-curved surface interaction point, are analytically analyzed, based on which the discrete constraints of the laser machining trajectory are determined accordingly. Secondly, a machining strategy of constant coincidence between the laser beam and the surface normal is proposed, for which a five-axis linkage motion micro-platform is designed and developed, and its machining performance is evaluated by developing a virtual simulation environment. Finally, the five-axis linkage laser micro-milling experiments and the characterization of as-fabricated complex curved surface microtextures are carried out, which demonstrate the realization of high-precision surface microtextures with complex shapes on stainless steel curved workpiece with high steepness and small size. The results suggest that the continuously and uniformly processing of complex curved surface microtextures can be realized by using the multi-axis linkage precision motion platform coupled with the nanosecond pulse laser beam, which provides a feasible method for preparing high-quality microtextures on complex surfaces with high steepness and small size. © 2024 Chinese Mechanical Engineering Society. All rights reserved.