In this talk, we will begin with the definition of a unimodular row and its relation to Serre's problem on projective modules. We 
 will then see under what conditions group structures exist on orbit spaces of unimodular rows under elementary action. Though the 
 group structure is quite complicated in general, there is a nice formula for the group structure in certain special cases like affine 
 algebras or polynomial rings. As a consequence of the nice group structure, one can complete certain unimodular rows elementarily too.
 The talk will be accessible to students.