In this study, we present a high-order numerical method based on a combined compact integrated RBF (IRBF) approximation for viscous flow and fluid structure interaction (FSI) problems. In the method, the fluid variables are locally approximated by using the combined compact IRBF, and the incompressible Navier-Stokes equations are solved by using the velocity-pressure formulation in a direct fully coupled approach. The fluid solver is verified through various problems including heat, Burgers, convection-diffusion equations, Taylor-Green vortex and lid driven cavity flows. It is then applied to simulate some FSI problems in which an elastic structure is immersed in a viscous incompressible fluid. For FSI simulations, we employ the immersed boundary framework using a regular Eulerian computational grid for the fluid mechanics together with a Lagrangian representation of the immersed boundary. For the immersed fibre/membrane FSI problems, although the order of accuracy of the present scheme is generally similar to FDM approaches reported in the literature, the present approach is nonetheless more accurate than FDM approaches at comparable grid spacings. The numerical results obtained by the present scheme are highly accurate or in good agreement with those reported in earlier studies of the same problems. (C) 2016 Elsevier Ltd. All rights reserved.