This invaluable volume is a collection of articles in memory of Jacques-Louis Lions, a leading mathematician and the founder of the contemporary French Applied Mathematics School. The contributions have been written by his friends, colleagues and students, including C Bardos, A Bensoussan, S S Chern, P G Ciarlet, R Glowinski, Gu Chaohao, B Malgrange, G Marchuk, O Pironneau, W Strauss, R Temam, etc. The book concerns many important results in analysis, geometry, numerical methods, fluid mechanics, control theory, etc.