$D$-meson and charmed-baryon measurement in $pp$ and $p$-Pb collisions with ALICE at the LHC

$A$ $Large$ $Ion$ $Collider$ $Experiment$ (ALICE) at CERN was designed to study the strongly interacting medium created in heavy-ion collisions at the LHC, the $Quark$-$Gluon$ $Plasma$ (QGP). Heavy quarks (charm and beauty), produced in the early stages of the collision, are among the most powerful probes to investigate QGP properties. On this regard, results from pp and p-Pb collisions constitute the needed reference in order to disentangle initial-state from QGP related effects.Charmed mesons ($\rm D^{0}$, $\rm D^{+}$, $\rm D^{*+}$ and $\rm D_{s}^{+}$) were reconstructed via their hadronic decays at mid-rapidity in pp collisions at $\sqrt{s}$ = 5.02, 7, 8 and 13 TeV and in p-Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV. The first measurement of the charmed baryon, $\Lambda_{\rm c}^{+}~$, at mid-rapidity was performed in pp collisions at $\sqrt{s}$ = 7 TeV and in p-Pb collisions at $\sqrt{s_{\rm NN}}~$ = 5.02 TeV through the full reconstruction of two of its hadronic decay channels and the partial reconstruction of one of its semileptonic decay channels. Moreover, the first measurement at the LHC of the $\rm \Xi_{c}^{0}~$$\rightarrow$ $e^{+}$$\rm \Xi^{-}$$\rm \nu_e$ production cross section was performed in pp collisions at $\sqrt{s}$ = 7 TeV. In this contribution recent results on charmed meson and baryon production, measured with the ALICE detector in pp and p-Pb collisions, will be presented.