Silicon carbide fibers having a high tensile strength are produced by (1) DISSOLVING AT LEAST ONE OF ORGANOSILICON HIGH MOLECULAR WEIGHT COMPOUNDS HAVING A SOFTENING POINT OF HIGHER THAN 50 DEG C, in which silicon and carbon are the main skeleton components, in a solvent for the organosilicon high molecular weight compounds or heating and melting said organosilicon high molecular weight compounds to prepare a spinning solution and spinning said spinning solution into filaments, (2) PRELIMINARILY HEATING THE SPUN FILAMENTS AT A TEMPERATURE OF 350 DEG -800 DEG C under vacuum to volatilize low molecular weight compounds contained in the high molecular weight compounds, and (3) BAKING THE THUS TREATED FILAMENTS AT A TEMPERATURE OF 800 DEG -2, 000 DEG C under vacuum or at least one non-oxidizing atmosphere selected from the group consisting of an inert gas, CO gas and hydrogen gas, to form silicon carbide fibers.