Abstract : The half wave Schrödinger equation is considered as a toy model motivated by the study of solutions to weakly dispersive and anisotropic equations. In this talk, we give an overview of results on the half wave Schrödinger equation, including well-posedness results, ill-posedness results, the ground state standing waves in the focusing case and the existence of modified wave operators in the defocusing case. We also introduce some open problems related to this equation.