Abstract: A curable composite material composition comprising (A) a polymerizable unsaturated compound, (B) a fiber reinforcing material and/or fiber, and (C) a polymerization initiator comprising an organic boron compound represented by the general formula (1) wherein R1, R2, R3 and R4 each independently represent an alkyl, aryl, allyl, aralkyl, alkenyl, alkynyl, silyl or heterocyclic group, a halogen atom, or a substituted alkyl, substituted aryl, substituted allyl, substituted aralkyl, substituted alkenyl, substituted alkynyl or substituted silyl group, and Z+ represents a cation, and an acidic compound, as well as a curing process for the curable composite material composition which includes irradiating and/or heating the curable composite material composition.