This book explores the most recent developments in the field of deterministic and stochastic fluid-structure interaction (FSI), which describes the coupled dynamical interaction between fluids and deformable structures. These sorts of problems arise in many real-life applications, including modeling blood flow in the human cardiovascular system, modeling bioartificial organs, and modeling large-scale structures such as wings, bridges, and dams. This work primarily focuses on the mathematical well-posedness of fluid-structure interaction (FSI) problems. It introduces a constructive theory in which solutions are built through a time-discretization approach based on operator-splitting strategies. This method has proven to be robust in analyzing FSI problems within both deterministic and probabilistic frameworks and can serve as a foundational framework for developing numerical schemes to effectively compute solutions to these highly complex multiphysics problems. As FSI is prevalent in science, a rigorous analysis of such coupled fluid-structure systems is key for continued technological development and progress in engineering. Consequently, this book can potentially benefit a broad range of readers, from advanced undergraduate and graduate students to researchers with a background in partial differential equations and fluid dynamics.