Let and be groups which act on themselves by conjugation and with a compatible action of on and of on . Then the non-Abelian tensor product is the group generated by the symbols subject to the relations
The authors in the present paper are mainly concerned with the computation of . Let be groups with given actions of on and and of and on . Under suitable conditions on these actions it is proved that . The tensor squares when is (i) the quaternion group of order ; (ii) the dihedral group of order ; (iii) the metacyclic group , , where and is odd; are computed. Another interesting result proved is that is the unique covering group of the perfect group . The tensor squares for non-Abelian groups of order obtained by using the Tietze transformation program are given. Also given are the generators and relations for for some of these groups. Some open problems are listed.