Longitudinal and transverse mixing in horizontal rotary kilns has been studied, using a three-dimensional Discrete Element Method approach. The focus is on the effect of the main operating conditions, i.e., the filling degree and the rotational speed of the rotary kiln, on quantitative measures of longitudinal and transverse mixing. Discrete Element Method simulations have been performed for various values of the main operating conditions. Flow regimes have been determined from visualisations of the simulations. It is shown that the mixing in the longitudinal direction can be described by a diffusion equation. The dependence of the diffusion coefficient on the operating conditions has been determined from the results of the simulations. It is found that the diffusion coefficient increases linearly with rotational speed, while the influence of the filling degree is relatively small. The mixing in the transverse plane is quantified by an entropy-like mixing index. The results of the simulations show that this mixing index varies exponentially with the number of revolutions of the rotary kiln. The dependence on the operating conditions of the transverse mixing speed, as determined from the exponential behaviour of the mixing index, has been determined. This mixing speed decreases with increasing rotational speed and with increasing filling degree. The transverse mixing speed that has been determined from the results of additional two-dimensional simulations is generally larger than that observed in the corresponding three-dimensional simulations. (c) 2005 Elsevier Ltd. All rights reserved.