The rupture speed of the earthquake fault rupture is a key parameter in geophysics and seismology researches. It is also an important factor in the strong ground motion modeling for earthquake engineering. In the past few decades, many studies have been focusing on or based on condition that the fault rupture speed be smaller than the local shear wave speed of the earth's crust. While with the development of modern broadband seismology and laboratory technique, evidences of earthquake fault rupture speed great than the local shear wave speed have been detected both in the real earthquakes observation and in laboratory experiments. In order to have an over-all view on earthquake super-shear rupture related studies, this paper summarized abundant of classical and state-of-the-art researches. Based on the study results, this paper firstly introduced the concepts of super-shear rupture, the limitation speed of crack rupture and then discussed some reasons that may cause a super-shear rupture. And then, studies on super-shear rupture are summarized in three aspects, including the real earthquake recording data based studies, the laboratory experiments studies, and theory and numerical modeling studies. In the end the some conclusions and discussions are given, it is suggested that the study for modeling of strong ground motion and characterization of ground motion under super-shear rupture are very essential fields to be noticed in the future works.