This paper reports the measurement of $\mathrm{J}/{\psi }$ meson production in proton–proton ( $\mathrm {p}\mathrm {p}$ ) and proton–lead ( $\mathrm {p}\mathrm {Pb}$ ) collisions at a center-of-mass energy per nucleon pair of $5.02,\text {TeV} $ by the CMS experiment at the LHC. The data samples used in the analysis correspond to integrated luminosities of 28 $,\text {pb}^{-1}$ and 35 $,\text {nb}^{-1}$ for $\mathrm {p}\mathrm {p}$ and $\mathrm {p}\mathrm {Pb}$ collisions, respectively. Prompt and nonprompt $\mathrm{J}/{\psi }$ mesons, the latter produced in the decay of ${\mathrm {B}}$ hadrons, are measured in their dimuon decay channels. Differential cross sections are measured in the transverse momentum range of $2