Invariance of means

We give a survey of results dealing with the problem of invariance of means which, for means of two variables, is expressed by the equality K. ( M, N) = K. At the very beginning the Gauss composition of means and its strict connection with the invariance problem is presented. Most of the reported research was done during the last two decades, when means theory became one of the most engaging and influential topics of the theory of functional equations. The main attention has been focused on quasi- arithmetic and weighted quasi- arithmetic means, also on some of their surroundings. Among other means of great importance Bajraktarevi ' c means and Cauchy means are discussed.