The half wave Schrödinger equation is considered as a toy model motivated by the study of solutions to weakly dispersive and anisotropic equations, with one direction corresponding to the half wave operator, which is not dispersive. In this talk, we give an overview of the results on the half wave Schrödinger equation, including the well-posedness results, the ill-posedness results, the ground state standing waves in the focusing case and the modified scattering results in the defocusing case.