  We study bounded ancient solutions of the Navier-Stokes equations. These are
the solutions which are defined for all past time. In two space dimensions we
prove that such solutions are either constant or functions of time only,
depending on the exact definition of admissible solutions. The general three
dimensional problem seems to be out of reach of existing techniques, but
partial results can be obtained in the case of axi-symmetric solutions. We
apply these results to some scenarios of potential singularity formation for
axi-symmetric solutions.
