This work presents a numerical technique for solving three-dimensional generalized Newtonian free surface flows. It is an extension to three dimensions of the technique introduced by Tome et al. [M.F. Tome, B. Duffy, S. McKee, A numerical technique for solving unsteady non-Newtonian free surface flows, J. Non-Newtonian Fluid Mech. 62 (1996) 9-34] but additionally includes many other features. The governing equations are solved by a finite difference method on a staggered grid. It uses marker particles to describe the fluid; these particles provide the location and visualization of the fluid free surface. As currently implemented, the present method can simulate generalized Newtonian flow in which the viscosity is modelled using the Cross model. The numerical technique presented in this paper is validated by using exact solutions for the flow of a Cross model fluid inside a pipe and convergence is demonstrated by means of grid refinement for the problem of a spreading drop. Numerical results showing the flow of a generalized Newtonian fluid jet impinging onto a flat surface and that of a jet buckling are given. (C) 2004 Elsevier B.V. All rights reserved.