The neural tube is an important model system of morphogenesis representing the developmental module of out-of-plane epithelial deformation. As the embryonic precursor of the central nervous system, the neural tube also holds keys to many defects and diseases. Recent advances begin to reveal how genetic, cellular and environmental mechanisms work in concert to ensure correct neural tube shape. A physical model is emerging where these factors converge at the regulation of the mechanical forces and properties within and around the tissue that drive tube formation towards completion. Here we review the dynamics and mechanics of neural tube morphogenesis and discuss the underlying cellular behaviours from the viewpoint of tissue mechanics. We will also highlight some of the conceptual and technical next steps.