This talk will discuss some algorithms for the efficient and accurate numerical solution of incompressible flows in complex moving geometry. The approach is based on high-order accurate approximations and boundary conditions as well as the use of composite overlapping grids. Overlapping grids are an efficient, flexible and accurate way to represent complex, possibly moving, geometry using a collection of overlapping structured grids. In the overlapping grid approach, narrow boundary fitted grids are located near boundaries and these grid move through stationary back-ground grids. For incompressible flows with moving geometry we have been developing a scheme that is based on an approximate-factored compact scheme for the momentum equations together with a matrix-free multigrid solver for the pressure equation. The overall scheme is fourth-order accurate in space, including boundaries, and currently second-order accurate in time. The scheme will be described and results will be presented for some three-dimensional (parallel) computations of flows with moving bodies.