“Topological Groups” are mathematical structures in which the notions of group and topology are blended. These partly geometric objects form a rich territory of interesting examples in topology and geometry, due to the presence of the two basic interrelated mathematical structures in one and the same set. The first section concerns with the basic properties and examples of topological groups. Here, we also deal with their separation properties. The notion of subgroups of a topological group is studied in the second section. In the third section, we treat quotient groups and isomorphisms theorems for topological groups. The notions of the decomposition of a topological group into direct product and semidirect product of its subgroups are discussed in the last section.