Starting pitches for carbon fibers

A process for making carbon fibers comprising heat treating a starting pitch to make a precursor pitch, melt spinning, infusibilizing, carbonizing or further graphitizing said precursor pitch to obtain carbon fibers wherein the starting pitch is produced by heat treating at 400.degree.-500.degree. C. under a hydrogen pressure of at least 20 Kg/cm.sup.2.G an oil selected from the group consisting of (1) a heavy fraction oil (A) boiling at not lower than 200.degree. C. obtained as a by-product at the time of fluidized catalytic cracking of petroleum and (2) a mixture of said heavy fraction oil (A) with a heavy fraction oil (B) boiling at not lower than 200.degree. C. obtained at the time of steam cracking of petroleum, the ratio by weight between said heavy fraction oil (B) to said heavy fraction oil (A) being 1 to 0.1-9.