The present invention provides a novel olefin polymerization catalyst having excellent olefin polymerization activity and, in polymerization with the catalyst comprising a transition metal compound, a polymerization process for preparing a polymer having a low molecular weight with high polymerization activity. The process comprises polymerizing olefin in the presence of an olefin polymerization catalyst comprising (A) a transition metal compound represented by the following ##STR00001## [M: a Group 4 or 5 transition metal atom in the Periodic Table, m: 1 4, R.sup.1: H, a C.sub.1 C.sub.5 linear hydrocarbon group, 3- to 5-membered alicyclic hydrocarbon group, bicycloaliphatic hydrocarbon group wherein two alicyclic rings share one or more carbon atoms, R.sup.2 to R.sup.6: H, a halogen atom, hydrocarbon group etc, X: H, a halogen atom etc, and n is a valence of M], and (B) at least one compound selected from (B-1) an organometallic compound, (B-2) an organoaluminum oxy compound and (B-3) an ionizing ionic compound.