Loewner theory for quasiconformal extensions: old and new

This survey article gives an account of quasiconformal extensions of univalent functions with its motivational background from Teichm\"uller theory and classical and modern approaches based on Loewner theory.Comment: 25 pages, 3 figs. This paper will be included in the proceedings of the 2nd GSIS-RCPAM International Symposium "Geometric Function Theory and Applications in Sendai" which was held in Tohoku University on September 10th-13th, 201