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The production cross sections of the prompt charmed mesons D0, D+, D∗+ and Ds were measured at mid-rapidity in p-Pb collisions at a centre-of-mass energy per nucleon pair sNN−−−√=5.02 TeV with the ALICE detector at the LHC. D mesons were reconstructed from their decays D0→K−π+, D+→K−π+π+, D∗+→D0π+, D+s→ϕπ+→K−K+π+, and their charge conjugates. The pT-differential production cross sections were measured at mid-rapidity in the interval 1<pT<24 GeV/c for D0, D+ and D∗+ mesons and in 2<pT<12 GeV/c for Ds mesons, using an analysis method based on the selection of decay topologies displaced from the interaction vertex. The production cross sections of the D0, D+ and D∗+ mesons were also measured in three pT intervals as a function of the rapidity ycms in the centre-of-mass system in −1.26<ycms<0.34. In addition, the prompt D0 cross section was measured in pp collisions at s√=7 TeV and p-Pb collisions at sNN−−−√=5.02 TeV down to pT=0 using an analysis technique that is based on the estimation and subtraction of the combinatorial background, without reconstruction of the D0 decay vertex. The nuclear modification factor RpPb(pT), defined as the ratio of the pT-differential D-meson cross section in p-Pb collisions and that in pp collisions scaled by the mass number of the Pb nucleus, was calculated for the four D-meson species and found to be compatible with unity within experimental uncertainties. The results are compared to theoretical calculations that include cold-nuclear-matter effects and to transport model calculations incorporating the interactions of charm quarks with an expanding deconfined medium.