The Benjamin-Ono equation is a dispersive PDE in one space dimension introduced by T.B. Benjamin in 1967 as a model for long internal gravity waves in a two-layer fluid with infinite depth. It admits a Lax pair involving Toeplitz operators on the Hardy space. I will explain how this structure leads to an explicit formula for the general solution in terms of its initial datum, in both cases of periodic boundary conditions and of solutions on the line decaying at infinity. Applications to low regularity wellposedness and to the zero-dispersion limit will be discussed.