We study the large-scale velocity fields traced by galaxy clusters in numerical simulations of a box of side 960 hs−1 Mpc, and compare them with available data on real clusters. In order to test the reliability of the simulations, which are based on an optimized version of the Zel’dovich approximation, we compare their cluster velocities with those of ‘exact’ N-body simulations, and find a remarkable agreement between the two according to a variety of statistical tests. We analyse cold dark matter (CDM) models with density parameter in the range 0.2 ⩽ Ω0 ⩽ 1, both with and without the cosmological constant term to provide a flat geometry. We also simulate a cold+hot dark matter (CHDM) model, with 30 per cent provided by the hot component. Comparison with real data is performed by applying tests based on the cumulative velocity frequency distribution (CVFD) and bulk flow statistics. For the CVFD, we use observational velocity data from different authors, and find that results based on different data sets are contradictory. In particular, the recent infrared Tully−Fisher (IRTF) data of Giovanelli yield smaller velocities with smaller errors than both the IRTF and Dns−σ data of Hudson. It turns out that the Giovanelli data are only only consistent with the open Ω0 = 0.4 and the flat Ω0 = 0.2 models, while the Hudson data, though less discriminatory because of their larger errors, appear to exclude open models with Ω0 ⩽ 0.4 and flat models with Ω0 = 0.2. This latter conclusion also holds if one pools all the data into a single sample regardless of the systematic differences in the two different sources. Furthermore, CVFD and bulk flow analyses of the Branchini et al. reconst-ructed velocity data again disfavour precisely those models accepted on the grounds of Giovanelli’s sample. Finally, we confirm that the Lauer & Postman reported bulk flow determination would be a rare event in the cosmological models we have analysed.