The phenomena of the breakup of liquid sheets and jets are encountered in nature as well as in various industrial applications. A good understanding of these phenomena requires a sound basic scientific knowledge of the dynamics of flows involving interfaces between different fluids. This book is the outcome of the author's inquiry into this fundamental knowledge. My understanding of the subject matter has been consolidated gradually through direct and indirect collaborations with my students and colleagues. The objective and scope of this book in the context of related existing works are explained in Chapter 1. Chapters 2 to 5 are devoted to exposition of the onset of sheet breakup. Chapters 6 to 10 discuss jet breakup. A perspective of the challenging aspects of the subject, including the nonlinear evolution subsequent to the onset of instability and nanojets, is sketched in Chapter 11. Some additional topics related to the breakup of a liquid body into smaller parts are discussed in the epilogue. Readers are expected to have the equivalent of at least an undergraduate background in science or engineering. In the theoretical development I have strived for mathematical rigor, numerical accuracy, and rational approximation. However, mathematics has not been used just for the sake of mathematics. I have depended on comparisons between different theories and experiments to establish physical concepts. Practical applications of the concepts are pointed out in appropriate places. The references relevant to each chapter are listed at the end of the chapter.