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Hadrons containing heavy quarks, i.e. charm and beauty, are effective probesto investigate the properties of the hot, dense and strongly-interacting mediumformed in high-energy nuclear collisions. The relatively large masses of heavyquarks ensure that they are predominantly produced in the early stages of thecollision and probe the complete space-time evolution of the expanding medium.The measurements of D-meson production in pp collisions provide an importanttest of pQCD calculations and serve as an essential baseline for thecomprehensive studies in heavy-ion collisions. The study of D-meson productionin p-Pb collisions is necessary to disentangle the cold nuclear matter effectsfrom hot nuclear matter effects. The measurement of heavy-flavour production asa function of charged-particle multiplicity in pp and p-Pb collisions couldprovide insight into the role of multi-parton interactions at LHC energies.  We present ALICE results on D-meson production in pp collisions at $\sqrt{s}=$ 7 TeV and p-Pb collisions at $\sqrt{s_{\rm {NN}}}=$ 5.02 TeV. The D-mesonyields per event, measured in different multiplicity intervals and normalizedto their multiplicity-integrated values, are presented for pp and p-Pbcollisions. The $p_{\rm {T}}$-differential production cross section and nuclearmodification factor of prompt D mesons are measured in p-Pb collisions. Thenuclear modification factor, $R_\mathrm{pPb}$, is compatible with unity withinuncertainties, indicating that cold nuclear matter effects are small for$p_{\mathrm{T}} \gtrsim 3$ GeV/$c$. The D-meson transverse momentumdistributions in p-Pb collisions relative to pp collisions, measured in severalmultiplicity classes, are also discussed.

Measurements of open-charm production in pp and p-Pb collisions with the ALICE detector at the LHC