In this book an efficient algorithm for simulation of deformable bodies interacting with incompressible flows is presented. The spatial discretization of the Navier-Stokes equations is based on compact finite differences. Using a uniform Cartesian grid we benefit from the advantage of a new fourth-order direct solver for the Poisson equation to ensure the incompressibility constraint down to machine zero over an optimal grid. For introducing a deformable body in fluid flow, the volume penalization method is used. The deformable body is interacting with the surrounding fluid due to the resulted hydrodynamic forces and the torque. An efficient law for controlling the curvature of an anguilliform fish, swimming toward a prescribed goal, is proposed which is based on the geometrically exact theory of nonlinear beams and quaternions. Furthermore to reduce the computational effort, adaptation of grid is performed by using multiresolution analysis. Finally an extension to three dimensional swimming is performed by adding the implicit volume penalization method to the Incompact3d open access code, to be able to take into account the deformable bodies’ interaction with incompressible flows.