The method of smoothed particle hydrodynamics (SPH) is extended to model incompressible flows of low Reynolds number. For such flows, modification of the standard SPH formalism is required to minimize errors associated with the use of a quasi-incompressible equation of state. Treatment of viscosity, state equation, kernel interpolation, and boundary conditions are described. Simulations using the method show close agreement with series solutions for Couette and Poiseuille flows. Furthermore, comparison with finite element solutions for flow past a regular lattice of cylinders shows close agreement for the velocity and pressure fields. The SPH results exhibit small pressure fluctuations near curved boundaries. Further improvements to the boundary conditions may be possible which will reduce these errors. A similar method to that used here may permit the simulation of other flows at low Reynolds numbers using SPH. Further development will be needed for cases involving free surfaces or substantially different equations of state.