These lecture notes cover a series of three two-hour lectures on fluid-structure interaction involving incompressible, viscous fluids, presented as the CIME Summer Workshop entitled “Progress in Mathematical Fluid Dynamics”, held in Cetraro, Italy, in June 2019. They are intended for graduate students and postdocs with interest in mathematical fluid dynamics. The goal was to present certain mathematical techniques, developed within the past 6 years, to study existence of weak solutions for a class of fluid-structure interaction problems between viscous, incompressible fluids and elastic, viscoelastic, or composite structures. The resulting problem is a nonlinear moving-boundary problem with a strong geometric nonlinearity associated with the fluid domain motion. The existence proof is constructive. It is based on semidiscretizing the coupled problem in time, also known as Rothe’s method, and then using a Lie operator splitting strategy to define fluid and structure subproblems, which communicate via the initial data. The Lie operator splitting is designed in just the right way so that the energy balance at the discrete level approximates well that of the coupled, continuous problem. To prove convergence of approximating sequences to a weak solution of the coupled problem, a recent generalization of the Aubin–Lions–Simon compactness lemma to problems on moving domains is used. The methodology presented here served as a basis for the construction of several loosely coupled partitioned schemes for solving fluid-structure interaction problems.