A butadiene polymer (i) having a cis bond unit content of at least 50% based on the total butadiene units, a number average molecular weight (Mn) of 1,000 to 10,000,000, and at least 80%, based on the total molecular chains, of living chains containing a transition metal of group IV of the Periodic table at a terminal thereof; a polymer (ii) obtained by modifying terminals of the polymer (i); and a polymer (iii) obtained by coupling the polymers (i). These polymers (i), (ii) and (iii) are obtained by polymerizing a conjugated diene monomer alone or with a copolymerizable monomer at a specific temperature in the presence of a catalyst comprising a compound (A) of a transition metal of group IV of the periodic table having a cyclopentadienyl structure and a co-catalyst (B) selected from organoaluminum-oxy compound (a) and others and optionally further by contacting the resultant polymer with a terminal modifier or a coupling agent.