This book presents a set of essays on interfacial instability, essays that are linked by a common question and a common focus: the question of the stability of a liquid jet and a focus on the surface tension acting at the interface between two phases. The discussion is analytic rather than numerical. The first two essays provide the necessary background; the remaining seven essays lead logically from these to the treatment of a series of applications, including inviscid and viscous fluids, the effects of rotation and the stability of freezing and melting fronts. The discussion is, for the most part, limited to problems that can be solved analytically, and thus to linear problems. The techniques used to analyze the problems can, however, be generalized to nonlinear problems and numerical methods. Intended for graduate students and researchers in fluid mechanics, the treatment assumes some knowledge of fluid mechanics. Students whose research problems involve multi-phase flows will find the concepts in this manuscript to be helpful in describing boundary conditions for their formulations.