There has been increasing interest in the study of flows in biological systems in recent years. Examples range from blood flow in the cardiovascular systems to flows past swimming animals and organisms. Such flows take place in multi-connected domains with complex geometries, flexible moving boundaries, over a range of Reynolds numbers, and usually involve non-linear fluid/structure interactions (FSI). An efficient and versatile computational framework for simulating a wide range of biological flows with complex moving boundaries and FSI has been developed. This computational framework is based on the hybrid Cartesian/immersed boundary method of Gilmanov and Sotiropoulos (Journal of computational physics, 2005) and the curvilinear/immersed boundary (CURVIB) method of Ge and Sotiropoulos (Journal of Computational Physics, 2007) This work has extended the above methods to non-linear FSI problems for multiple, arbitrarily complex bodies in a general non-inertial reference frame (CURVIB-FSI solver).;The stability of the CURVIB-FSI computational framework for different FSI coupling methods has been analyzed by some simple, albeit similar, FSI problems and the behaviors of the FSI solver in the numerical experiments were explained. It was shown that the ratio of the added mass to the mass of the structure as well as the sign of the local time rate of change of the force or moment imparted on the structure by the fluid determine the stability and convergence of the FSI algorithm. The stabilizing role of under-relaxation is also clarified and the upper bound of the under-relaxation coefficient required for stability is derived.;The CURVIB-FSI solver has been validated in both inertial and non-inertial reference frames by applying it to several well-studied FSI problem e.g. vortex-induced vibrations of an elastically mounted cylinder. After demonstrating the accuracy and validity of the CURVIB-FSI, it is applied to simulate three important biological flows: (1) Flow through a bi-leaflet mechanical heart valves; (2) Aquatic fishlike swimming; (3) Aquatic planktonic swimming. Each of these simulations is the most advanced in their field and pushes the limits of the state-of-the-art numerical simulations. Furthermore, new insights have been gained into the physics of these important flows, which with experiments alone would not have been possible.