Understanding human organ formation is a scientific challenge with far-reaching medical implications^1,2. Three-dimensional stem-cell cultures have provided insights into human cell differentiation^3,4. However, current approaches use scaffold-free stem-cell aggregates, which develop non-reproducible tissue shapes and variable cell-fate patterns. This limits their capacity to recapitulate organ formation. Here we present a chip-based culture system that enables self-organization of micropatterned stem cells into precise three-dimensional cell-fate patterns and organ shapes. We use this system to recreate neural tube folding from human stem cells in a dish. Upon neural induction^5,6, neural ectoderm folds into a millimetre-long neural tube covered with non-neural ectoderm. Folding occurs at 90% fidelity, and anatomically resembles the developing human neural tube. We find that neural and non-neural ectoderm are necessary and sufficient for folding morphogenesis. We identify two mechanisms drive folding: (1) apical contraction of neural ectoderm, and (2) basal adhesion mediated via extracellular matrix synthesis by non-neural ectoderm. Targeting these two mechanisms using drugs leads to morphological defects similar to neural tube defects. Finally, we show that neural tissue width determines neural tube shape, suggesting that morphology along the anterior-posterior axis depends on neural ectoderm geometry in addition to molecular gradients⁷. Our approach provides a new route to the study of human organ morphogenesis in health and disease.