When the peak power of the pulse satisfies certain conditions, it will cause the transverse intensity distribution at the output end of the gradient refractive index multimode fiber to transform from speckle to bell shaped beam. This is a special nonlinear phenomenon caused by the spatiotemporal nonlinear interaction between multiple transverse modes, known as Kerr beam self-cleaning (BSC). When this process occurs, it exhibits a strong mode self purification process: the energy of high-order modes is transferred to the fundamental mode, that is, the energy exchange between modes. At this time, the energy of high-order modes continuously transfers to the fundamental mode and is stably stored in the fundamental mode. Its significant characteristics are: improving the brightness of the output beam and robustness to fiber bending and compression.These features enable us to overcome the limitations of multimode fibers in terms of low output beam quality, which is extremely promising for a host of technological applications. We introduced this phenomenon from its discovery to recent research progress, and summarized its mechanism and explanation.