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Problems in biological fluid dynamics typically involve the interaction of an elastic structure with its surrounding fluid. A unified computational approach, based on an immersed boundary framework, couples the internal force-generating mechanisms of organisms and cells with an external, viscous, incompressible fluid. Computational simulation, in conjunction with laboratory experiment, can provide valuable insight into complex biological systems that involve the interaction of an elastic structure with a viscous, incompressible fluid. This biological fluid-dynamics setting presents several more challenges than those traditionally faced in computational fluid dynamics - specifically, dynamic flow situations dominate, and capturing time-dependent geometries with large structural deformations is necessary. In addition, the shape of the elastic structures is not preset: fluid dynamics determines it. This article presents our recent progress on coupling the internal molecular motor mechanisms of beating cilia and flagella with an external fluid, as well as the three-dimensional (3D) undulatory swimming of nematodes and leeches. We expect these computational models to provide a testbed for examining different theories of internal force-generation mechanisms.