Singularities of the Navier-Stokes equations occur when some derivative of the velocity field is infinite at any point of a field of flow (or, in an evolving flow, becomes infinite at any point within a finite time). Such singularities can be mathematical (as e.g. in two-dimensional flow near a sharp corner, or the collapse of a Mobius-strip soap film onto a wire boundary) in which case they can be resolved by refining the geometrical description; or they can be physical (as e.g. in the case of cusp singularities at a fluid/fluid interface) in which case resolution of the singularity involves incorporation of additional physical effects; these examples will be briefly reviewed. The finite-time singularity problem for the Navier-Stokes equations will then be discussed and a recently developed analytical approach will be presented; here it will be shown that, even when viscous vortex reconnection is taken into account, there is indeed a physical singularity, in that, at sufficiently high Reynolds number, vorticity can be amplified by an arbitrarily large factor in an extremely small point-neighbourhood within a finite time, and this behaviour is not resolved by viscosity. Similarities with the soap-film-collapse and free-surface--cusping problems are noted in the concluding section, and the implications for turbulence are considered.

8 pages, 8 figures, to appear in Phys. Rev. Fluids 2019