In recent decades, developments in non-linear mathematics and electronic computers have led to substantial advances in the field of fluid mechanics and related transport phenomena. However, many questions remain: How do two or three fluid phases move together? What are the relationships between flows in porous and fractured materials? These and similar questions provide opportunities for synergy. This book explores intriguing and difficult mathematical problems with the hope that mathematical and computational advances will increase practitioners' understanding and result in better efficiency and lower costs for oil and gas production.