This paper presents a local moving least square-one-dimensional integrated radial basis function networks method for solving incompressible viscous flow problems using stream function-vorticity formulation. In this method, the partition of unity method is employed as a framework to incorporate the moving least square and one-dimensional integrated radial basis function networks techniques. The major advantages of the proposed method include the following: (i) a banded sparse system matrix which helps reduce the computational cost; (ii) the Kronecker- δ property of the constructed shape function which helps impose the essential boundary condition in an exact manner; and (iii) high accuracy and fast convergence rate owing to the use of integration instead of conventional differentiation to construct the local radial basis function approximations. Several examples including two-dimensional (2D) Poisson problems, lid-driven cavity flow and flow past a circular cylinder are considered, and the present results are compared with the exact solutions and numerical results from other methods in the literature to demonstrate the attractiveness of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.