In this paper we will review the fundamental aspects of the supershear earthquake ruptures, i.e., events which propagate with speeds greater than the shear wave speed of the medium surrounding the fault. The rupture starts from an imposed hypocenter and then it spreads spontaneously (i.e., without imposed rupture speed) over the entire fault plane. The latter is characterized by a rheology described by the linear slip-weakening model, which analytically prescribe the evolution of the fault frictional as a function of the cumulated fault slip. The problem is solved numerically, by using two different finite-difference schemes, because it is not possible, even in homogeneous conditions, to solve the elastodynamic problem in an analytical, closed form. We will discuss the main differences exsisting between sub- and supershear synthetic earthquakes, by considering both the on-fault and the off-fault solutions. The results presented here summarizes the most prominent and recent researches in this field, as well as the seismological implications.