The topology of free topological groups

The known explicit descriptions of the topology of free and free Abelian topological groups are collected. Examples of the application of such descriptions by various authors to study the properties of topological groups and some related topological-algebraic objects are given. Various approaches to describing the topology of free topological groups are presented and a general method for topologizing free groups is suggested. The fundamental properties of free topological groups and their structure are considered. The most important results on free topological groups obtained by various authors through the years are mentioned.