A topological group is an abstract group which is also a topological space and in which the group operation are continuous. In group theory the algebraic binary operation of passage to a limit is studied in a similar manner. The two fundamental mathematical concepts of binary operation and passage to a limit are united and interrelated in the concept of topological group. The concept of topological groups arose from the study of continuous transformations. However, topological groups can be studied quite independently from continuous transformations and the latter can be presented as applications of topological groups. The first person to consider topological groups was Lie, but he was concerned with groups defined by analytic operations. Later, around 1900-1910 other men, beginning with Hilbert and Brouwer, studied more general topological groups. The topological group is then -- from a logical point of view only -- a combination of the abstract group and the topological space. Hence, the first and second chapters of this paper will be devoted to the concept of abstract group and topological space respectively, while the third and main chapter will utilize these two concepts in the formation and study of the topological group. Our main source of information will be Leon Pontrjagin's book, "Topological Groups" (1939); however, our approach will be somewhat broader and we will include results from other sources and our own investigations. In order to avoid making this paper to lengthy for its purpose, we will prove only some of the theorems. The rest of them will be simply stated and often followed by a sketch or an outline of the proof. The major definitions and theorems and all the examples will be numbered consecutively as they appear. For instance "Theorem 2.5" is the fifth numbered item in the second chapter.