  We present an overview of the ALICE measurements on the production of
$\mathrm{J/}\psi$ in Pb-Pb collisions at a centre-of-mass energy per nucleon
pair of $\sqrt{s_\mathrm{NN}} = 2.76\,\mathrm{TeV}$ and p-Pb collisions at
$\sqrt{s_\mathrm{NN}}= 5.02\,\mathrm{TeV}$ at mid-rapidity ($|y_\mathrm{lab} |
< 0.8$) down to zero transverse momentum. The cold nuclear matter effects
estimated from the p-Pb measurements and their impact on the interpretation of
Pb-Pb results are discussed, based on comparison of data to model calculations.
