The groups of homeomorphisms of the real line and circle are universal object of the groups theory. On the other hand, at study of many problems of geometry, the functions theory of the complex variable, the dynamic systems which are reduced to systems of small dimension, the basic obstacle is been the structure of the group of homeomorphisms of real line (circle), which is generated canonical image by initial problem. Therefore, the study of structure of such groups is actual. The report is devoted to investigation of the major characteristics of the group of homeomorphisms of the real line (circle). Topological, metric, algebraic and combinatory characteristics are concerned to them. The basic part of my report will be devoted to correlations between characteristics having a various nature.