A catalyst for olefin polymerization which comprises (A) a transition metal compound according to the general formula (I) below, (B) a compound capable of reacting with the component (A) to form an ionic complex, and (C) an organic aluminum compound, where the general formula (I) is ##STR52## wherein M is Ti, Zr or Hf, E.sup.1 and E.sup.2 are each a cyclopentadienyl group, a substituted cyclopentadienyl group, an indenyl group or substituted indenyl group, and they form a cross-linking structure via (A.sup.1).sub.p, (A.sup.2).sub.p, . . . (A.sup.n).sub.p and (D).sub.s, and they may be the same or different;
- X is a .sigma.-bonding ligand, and when a plurality of Xs are present, these plural Xs may be the same or different and each X may cross-link with E.sup.1, E.sup.2 or L;
- L is a Lewis base, and when a plurality of Ls are present, these plural Ls may be the same or different, and each L may cross-link with E.sup.1, E.sup.2 or X;
- A.sup.1, A.sup.2, . . . A.sup.n are each a crosslinking group consisting of at least one element which is C, H, Si, O, N, S or P and they may be the same or different, but at least one of them comprises a crosslink consisting of carbon alone;
- D is a crosslinking group consisting of at least one element which is C, H, Si, O, N or P, and when a plurality of Ds are present, these plural Ds may be the same or different;
- n is an integer of 2 to 4;
- p is an integer of 1 to 4, and the respective ps may be the same or different;
- q is an integer of 1 to 5 and is (the valence of M)-2;
- r is an integer of 0 to 3;
- and s is an integer of 0 to 4, and when s is 0, (A.sup.1).sub.p, (A.sup.2).sub.p, . . . (A.sup.n).sub.p and E.sup.2 form a direct bond.