We use the Chandra X-ray Observatory to study the dark matter haloes of 34 massive, dynamically relaxed galaxy clusters, spanning the redshift range 0.06 < z < 0.7. The observed dark matter and total mass (dark-plus-luminous matter) profiles can be approximated by the Navarro–Frenk–White (hereafter NFW) model for cold dark matter (CDM) haloes; for ∼80 per cent of the clusters, the NFW model provides a statistically acceptable fit. In contrast, the singular isothermal sphere model can, in almost every case, be completely ruled out. We observe a well-defined mass–concentration relation for the clusters with an intrinsic scatter in good agreement with the predictions from simulations. The slope of the mass–concentration relation, c∝Mavir/(1 +z)b with a=−0.45 ± 0.12 at 95 per cent confidence, is steeper than the value a∼− 0.1 predicted by CDM simulations for lower mass haloes. With the slope a included as a free fit parameter, the redshift evolution of the concentration parameter, b= 0.71 ± 0.52 at 95 per cent confidence, is consistent with the same simulations (b∼ 1). Fixing a∼−0.1 leads to an apparent evolution that is significantly slower, b= 0.30 ± 0.49, although the goodness of fit in this case is significantly worse. Using a generalized NFW model, we find the inner dark matter density slope, α, to be consistent with unity at 95 per cent confidence for the majority of clusters. Combining the results for all clusters for which the generalized NFW model provides a good description of the data, we measure α= 0.88 ± 0.29 at 95 per cent confidence, in agreement with CDM model predictions.