Fluid-structure interaction (FSI) is ubiquitous in nature and occurs at molecular to environmental scales, from the writhing of DNA in nucleoplasm, to the beating of cilia and flagella and the projection of lamellipodia and bleb-like protrusions by motile cells, to the flow of blood in the heart, to swimming fish and flying birds and insects, to the dispersal of seeds and pollen in the wind. This talk will describe numerical methods and computational infrastructure for FSI, focusing on extensions of the immersed boundary (IB) method for fluid-structure interaction and applications of these methods to various models in medicine and biology. Different approaches are needed for FSI involving rigid and elastic structures, but both can be addressed within the framework of the IB method. I will discuss IB methods for FSI with prescribed structural kinematics and methods for FSI involving flexible bodies that use nonlinear structural dynamics formulations. I also will describe new extensions of these methods that aim to achieve higher-order accuracy for applications involving realistic biological and physiological geometries. I will survey applications of these IB methods in biology and medicine, including flagellar mechanics, aquatic locomotion and neuro-mechanical feedback, and esophageal transport. I will also detail ongoing work to develop IB models of the heart and its valves and applications to cardiovascular medical devices.