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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 Bajraktarević means and Cauchy means are discussed.

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Let I ⊂ R be an open interval and p, q, r ∈ (0, 1). We find all continuous and strictly monotonic functions α, β, γ : I → R satisfying the functional equationgeneralizing the Matkowski-Sutô equation. In the proof we adopt a method elaborated by Z. Daróczy and Zs. Páles when solving the Matkowski-Sutô equation, some results of A. Járai on improving regularity of solutions and an extension theorem by Z. Daróczy and G. Hajdu. We also use a theorem giving the form of all twice continuously differentiable solutions of the above equation proved jointly with J. Matkowski.