In this talk I will discuss well-posedness results of the Benjamin-Ono (BO) equation on the torus in the Sobolev

spaces $H^{s}$ for $s > - 1/2$ with $s = -1/2$ being the critical Sobolev exponant. The result is sharp

in the sense that the BO solution map does not extend continuously to $H^{s}$ for $s \le - 1/2$.

Our method of proof can also be used to analyze regularity properties of the solution map and to show

that the constructed solutions are almost periodic in time and that their orbits are relatively compact.

This is joint work with Patrick G\'erard and Peter Topalov. T.K. partially supported by the Swiss National Science

Foundation (project number 200020-165537). P.T. partially supported by the Simons Foundation (award number 526907).