The Galerkin boundary node method (GBNM) is a boundary only meshless method that combines an equivalent variational formulation of boundary integral equations for governing equations and the moving least-squares (MLS) approximations for generating the trial and test functions. In this approach, boundary conditions can be implemented directly and easily despite of the fact that the MLS shape functions lack the delta function property. Besides, the resulting formulation inherits the symmetry and positive definiteness of the variational problems. The GBNM is developed in this paper for solving three-dimensional stationary incompressible Stokes flows in primitive variables. The numerical scheme is based on variational formulations for the first-kind integral equations, which are valid for both interior and exterior problems simultaneously. A rigorous error analysis and convergence study of the method for both the velocity and the pressure is presented in Sobolev spaces. The capability of the method is also illustrated and assessed through some selected numerical examples. Copyright © 2011 John Wiley & Sons, Ltd.