As one of the core areas in the studies of fluid flows, interfacial dynamics has been attracting more and more attentions over the past few years. The interest of studying the behaviors of fluids near where they meet is interdisciplinary: many such problems arise from physics, chemical engineering, material science and applied mathematics. This book focuses on two mathematical approaches to the interfacial problems: conformal mapping methods and variational methods ? the former provides a strong connection between holomorphic functions and incompressible and irrotational flows, and the latter can be used to derive conditions which a deformable object in contact with fluids must satisfy. Three different problems in fluid dynamics are analyzed, each of which is motivated by experimental observations. Graduate students and senior undergraduate students in mathematics, physics or engineering who intend to undertake researches in fluid dynamics should find this book useful.