  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 $\sqrt{s_{\rm NN}}$ = 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
($p_{\rm T}$) 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 $p_{\rm T}$ = 0, which allowed a
model-independent determination of the $p_{\rm T}$-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 ($R_{\rm AA}$) of prompt D mesons at
$p_{\rm T}$ = 6-8 GeV/$c$ for the 0-10% and 30-50% centrality classes,
respectively. The D-meson $R_{\rm AA}$ is compared with that of charged pions,
charged hadrons, and J/$\psi$ mesons as well as with theoretical predictions.
The analysis of the agreement between the measured $R_{\rm AA}$, 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 $R_{\rm AA}$ 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.
