The production of prompt D⁰, D⁺, and D^*+ mesons was measured at midrapidity (|y| < 0.5) in Pb–Pb collisions at the centre-of-mass energy per nucleon–nucleon pair sNN = 5.02 TeV with the ALICE detector at the LHC. The D mesons were reconstructed via their hadronic decay channels and their production yields were measured in central (0–10%) and semicentral (30–50%) collisions. The measurement was performed up to a transverse momentum (p_T) of 36 or 50 GeV/c depending on the D meson species and the centrality interval. For the first time in Pb–Pb collisions at the LHC, the yield of D⁰ mesons was measured down to p_T = 0, which allowed a model-independent determination of the p_T-integrated yield per unit of rapidity (dN/dy). A maximum suppression by a factor 5 and 2.5 was observed with the nuclear modification factor (R_AA) of prompt D mesons at p_T = 6–8 GeV/c for the 0–10% and 30–50% centrality classes, respectively. The D-meson RAA is compared with that of charged pions, charged hadrons, and J/ψ mesons as well as with theoretical predictions. The analysis of the agreement between the measured R_AA, elliptic (v₂) and triangular (v₃) flow, and the model predictions allowed us to constrain the charm spatial diffusion coefficient D_s. Furthermore the comparison of R_AA and v₂ with different implementations of the same models provides an important insight into the role of radiative energy loss as well as charm quark recombination in the hadronisation mechanisms. [Figure not available: see fulltext.]