Abstract: In this paper we give a short survey on a problem on extremal quasiconformal extensions. It had been a conjecture for a long time that the dilatations K₀(h) and K₁(h) are equal before Anderson and Hinkkanen disproved this by constructing concrete examples of a family of affine mappings of some parallelograms. The problem also engendered many interesting results. At the end of the current paper, we discuss relationships among K₀(h), H(h) and K₁(h) as a concluding remark.

Key words: Quasisymmetric mapping, Extremal quasiconformal mapping, Universal Teichmüller space, Non-Strebel point

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