We describe the corkscrew point spread function (PSF), which can localize objects in three dimensions throughout a $3.2\mathrm{\mu m}$ depth of field with nanometer precision. The corkscrew PSF rotates as a function of the axial (z) position of an emitter. Fisher information calculations show that the corkscrew PSF can achieve nanometer localization precision with limited numbers of photons. We demonstrate three-dimensional super-resolution microscopy with the corkscrew PSF by imaging beads on the surface of a triangular polydimethylsiloxane (PDMS) grating. With 99,000 photons detected, the corkscrew PSF achieves a localization precision of $2.7\mathrm{nm}$ in x, $2.1\mathrm{nm}$ in y, and $5.7\mathrm{nm}$ in z.