We present a 90 per cent flux-complete sample of the 201 X-ray-brightest clusters of galaxies in the northern hemisphere (δ ≥ 0°), at high Galactic latitudes (ǀbǀ ≥ 20°), with measured redshifts z ≤ 0.3 and fluxes higher than 4.4 × 10−12 erg cm−2 s−1 in the 0.1–2.4 keV band. The sample, called the ROSAT Brightest Cluster Sample (BCS), is selected from ROSAT All-Sky Survey data and is the largest X-ray-selected cluster sample compiled to date. In addition to Abell clusters, which form the bulk of the sample, the BCS also contains the X-ray-brightest Zwicky clusters and other clusters selected from their X-ray properties alone. Effort has been made to ensure the highest possible completeness of the sample and the smallest possible contamination by non-cluster X-ray sources. X-ray fluxes are computed using an algorithm tailored for the detection and characterization of X-ray emission from galaxy clusters. These fluxes are accurate to better than 15 per cent (mean 1σ error). We find the cumulative log N—log S distribution of clusters to follow a power law κ S−α with α = 1.31+0.06−0.03 (errors are the 10th and 90th percentiles) down to fluxes of 2 × 10−12 erg cm−2 s−1, i.e. considerably below the BCS flux limit. Although our best-fitting slope disagrees formally with the canonical value of −1.5 for a Euclidean distribution, the BCS log N—log S distribution is consistent with a non-evolving cluster population if cosmological effects are taken into account. Our sample will allow us to examine large-scale structure in the northern hemisphere, determine the spatial cluster—cluster correlation function, investigate correlations between the X-ray and optical properties of the clusters, establish the X-ray luminosity function for galaxy clusters, and discuss the implications of the results for cluster evolution.