Layer-by-layer additive manufacturing is commonly utilized for additive manufacturing. Recent works utilize curved layers (rather than planar ones), on which print-paths are located, and outline their advantage over planar slicing. In this paper, free-form three-dimensional curves are utilized as input for the generation of print-paths, which covers the model to be printed and do not necessarily lie on either a planar or a curved layer. Such print-paths have been recently studied for 3-axis additive manufacturing, and a novel additive manufacturing process for the models represented using such curves are proposed for 4 and 5-axis additive manufacturing in this paper. The input curves are first subdivided into short sub-curves (i.e., segments), which are then merged to obtain print-paths with (collision-free) printing-head orientations along them. Thanks to additional two rota-tional axes of the printing-head, a less number of print-paths can potentially be obtained, which can reduce subdivisions in the input curves, and therefore, is desirable in additive manufacturing for improved mechanical properties in the printed parts. As a proof of concept, the print-paths with printing-head orientations along them are finally validated using an AM simulator and machine.