We present the charged-particle pseudorapidity density in Pb–Pb collisions at s_NN=5.02 TeV in centrality classes measured by ALICE. The measurement covers a wide pseudorapidity range from −3.5 to 5, which is sufficient for reliable estimates of the total number of charged particles produced in the collisions. For the most central (0–5%) collisions we find 21400±1300, while for the most peripheral (80–90%) we find 230±38. This corresponds to an increase of (27±4)% over the results at s_NN=2.76 TeV previously reported by ALICE. The energy dependence of the total number of charged particles produced in heavy-ion collisions is found to obey a modified power-law like behaviour. The charged-particle pseudorapidity density of the most central collisions is compared to model calculations — none of which fully describes the measured distribution. We also present an estimate of the rapidity density of charged particles. The width of that distribution is found to exhibit a remarkable proportionality to the beam rapidity, independent of the collision energy from the top SPS to LHC energies.