During development coordinated cell behaviors orchestrate tissue and organ morphogenesis to suit the lifestyle of the organism. We have used here the crustacean Parhyale hawaiensis to study the cellular basis of limb development. Transgenic Parhyale embryos with fluorescently labeled nuclei were imaged at high spatiotemporal resolution with multi-view light-sheet fluorescence microscopy over several days of embryogenesis spanning appendage morphogenesis from early specification up to late differentiation stages. Cell tracking with a new tool called Massive Multi-view Tracker (MaMuT) enabled the reconstruction of the complete cell lineage of an outgrowing thoracic limb with single-cell resolution. In silico clonal analyses suggested that the limb primordium becomes subdivided from an early stage first into anterior-posterior and then into dorsal-ventral compartments whose boundaries intersect at the distal tip of the growing limb. Limb bud formation is associated with the spatial modulation of cell proliferation, while limb elongation is also driven by the preferential orientation of division of epidermal cells along the proximal-distal axis of growth. Cellular reconstructions were predictive of the expression patterns of limb development genes including the Decapentaplegic (Dpp) morphogen.