Fluid–structure interaction (FSI) problems commonly encountered in engineering, environmental and biological applications involve geometrically complex flexible or rigid bodies undergoing large deformations. Immersed boundary (IB) methods have emerged as a powerful simulation tool for tackling such flows due to their inherent ability to handle arbitrarily complex bodies without the need for expensive and cumbersome dynamic re-meshing strategies. Depending on the approach such methods adopt to satisfy boundary conditions on solid surfaces they can be broadly classified as diffused and sharp interface methods. In this talk I will outline a powerful sharp-interface IB computational framework we have developed in my group for simulating complex FSI problems encountered in a broad range of applications in engineering and biology. I will demonstrate the capabilities of the method by presenting simulation results from various problems, including vortex-induced vibrations, aquatic swimming, cardiovascular fluid mechanics, river flows over mobile sediment beds, and wind and marine and hydrokinetic energy.