A bstract The production of prompt D 0 , 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 s NN = 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 D 0 mesons was measured down to p T = 0, which allowed a model-independent determination of the p T -integrated yield per unit of rapidity (d N/ d y ). 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 R AA 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 ( v 2 ) and triangular ( v 3 ) flow, and the model predictions allowed us to constrain the charm spatial diffusion coefficient D s . Furthermore the comparison of R AA and v 2 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.