Superconducting quantum interference devices (SQUIDs) can be used to detect weak magnetic fields and have traditionally been the most sensitive magnetometers available. However, because of their relatively large effective size (on the order of 1 mu m)(1-4), the devices have so far been unable to achieve the level of sensitivity required to detect the field generated by the spin magnetic moment (mu(B)) of a single electron(5,6). Here we show that nanoscale SQUIDs with diameters as small as 46 nm can be fabricated on the apex of a sharp tip. The nano-SQUIDs have an extremely low flux noise of 50 n Phi(0) Hz(-1/2) and a spin sensitivity of down to 0.38 mu(B) Hz(-1/2), which is almost two orders of magnitude better than previous devices(2,3,7,8). They can also operate over a wide range of magnetic fields, providing a sensitivity of 0.6 mu(B) Hz(-1/2) at 1 T. The unique geometry of our nano-SQUIDs makes them well suited to scanning probe microscopy, and we use the devices to image vortices in a type II superconductor, spaced 120 nm apart, and to record magnetic fields due to alternating currents down to 50 nT.