A variety of topological groups is a class of (not necessarily Hausdorff) topological groups closed under the operations of forming subgroups, quotient groups and arbitrary products. The variety of topological groups generated by a class of topological groups is the smallest variety containing the class. In this paper methods are presented to distinguish a number of significant varieties of abelian topological groups, including the varieties generated by (i) the class of all locally compact abelian groups;
(ii) the class of all k_omega-groups; (iii) the class of all sigma-compact groups; and (iv) the free abelian topological group on [0,1]. In all cases, hierarchical containments are determined.