Since the appearance of extremal rays and the minimal model program around 1980, we have seen the tremendous development of Algebraic Geometry. With this in mind, the conference 'Minimal Models and Extremal Rays' was held at RIMS Kyoto University in June 2011. The purpose was to review the past, examine the present and enjoy discussing the future. This volume is the outcome of this conference and consists of one survey article on the mathematical work of Professor Shigefumi Mori, who turned sixty in 2011, and thirteen research papers given by the authors: L Ein-S Ishii-M Mustata, J Kollár, S Kovács-K Schwede, D Greb-S Kebekus-T Peternell, H Kawanoue-K Matsuki, A-S Kaloghiros-A Kuronya-V Lazic, J-M Hwang-K Oguiso, M Lehn-Y Namikawa-Ch Sorger-D van Straten, O Fujino, J Chen, C Hacon-C Xu, S Mukai, Y Prokhorov-M Reid.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets except North America