Heavy-flavour production in small systems with ALICE at the LHC

Heavy quarks are produced in the initial stages of hadronic collisions via hard scattering processes and experience the complete evolution of the hot and dense medium created in heavy-ion collision, known as quark-gluon plasma. Measurement of their production in pp collisions provides baseline for observations of hot-medium effects in heavy-ion collisions, as well as tests of perturbative QCD calculations. A deeper understanding of heavy-flavour production in Pb$-$Pb collisions requires a detailed study of Cold Nuclear Matter effects in order to disentangle the role of initial- and final-state effects on their production. Measurements in small systems at high multiplicity recently gained additional interest due to the observation of the signal typical of the heavy-ion phenomenology. To study in detail the (initial- and/or final-state) effects playing a role on these observations, measurement of heavy-flavour production as a function of multiplicity is performed. In this talk, the most recent results on production cross section, nuclear modification factor and multiplicity dependence studies of production of D mesons and heavy-flavour hadron decay electrons at mid-rapidity and of muons at forward rapidity will be presented. The results will be compared with model calculations.