A novel composite superconductor includes a superconducting material and a normal conducting material. The composite superconductor has a number of closely spaced grooves on a surface being in contact with a refrigerant, and a number of ridges spaced apart from each other by the grooves and having acutely tapered ends. The surfaces of the grooves and the ridges are coated with a thin film of metal or its compound having higher heat resistance than that of the surface of the composite superconductor. Consequently, a high critical heat flux in film boiling and a high cooling capability can be provided, thereby making it possible to decrease the sectional area of the normal conducting material, to increase the effective current density and to miniaturize the size of a magnet.