In this work, several low-order models are derived to describe and simulate fluid-structure interaction problems with rigid bodies at a modest computational cost. The models are based on the inviscid flow assumption such that potential theory can be used with, in some cases, point vortices in the flow. Three general areas of application are considered. First, a thin airfoil undergoing small-scale unsteady motions in the presence of a freestream flow is investigated. The low-order model that is developed has only one ordinary differential equation for the fluid dynamic variables. This model is used to briefly investigate vortex-induced flutter in the attached-flow regime and control of a free-flying airfoil using synthetic jet actuators. Second, the vortex-induced vibrations of an arbitrary bluff body in the presence of vortices, with or without a freestream flow, are considered. Several examples of the canonical mass-spring-damper system for a circular cylinder and a flat plate are given to demonstrate the use of the vortex-based model for these applications. Finally, the two-body problem in a potential flow is addressed. A relatively simple solution specific to the doubly connected domain is determined and its resulting force and moment are coupled to the rigid bodies to investigate the mutual interactions between the two bodies. Aspects of drafting behind a forced body, the role of the fluid in elastic collision, and flapping flight are discussed in this context. Although a few specific examples and applications are given for each chapter, the main purpose of the thesis is to present low-order potential flow methods that are applicable to a variety of situations.