  The photoproduction of heavy vector mesons in the electromagnetic
interactions of ultra-relativistic nuclei is sensitive to the gluon
distribution in the nucleus and thus to cold nuclear matter effects like
shadowing or parton saturation. Besides the well known observations of vector
meson production in ultra-peripheral collisions, first observations of an
excess over the expected hadronic J/$\psi$ production at very low transverse
momentum ($p_T < $~0.3 GeV/$c$) in peripheral and semi-central nucleus-nucleus
collisions both at LHC and RHIC energies were interpreted as the first sign of
coherent J/$\psi$ photoproduction occurring in Pb-Pb collisions with nuclear
overlap. The ALICE Collaboration published the J/$\psi$ coherent
photoproduction cross sections in peripheral and semi-central Pb-Pb collisions
at $\sqrt{s_{\rm NN}}$ = 2.76~TeV and forward rapidity ($2.5<y<4.0$). Using the
LHC Run-2 data, ALICE presents preliminary results in peripheral Pb-Pb
collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV at mid-rapidity ($|y|<0.9$) and
forward rapidity. Thanks to the very good tracking resolution of the central
barrel, the extraction of the $p_T$-differential cross section was also
possible, strengthening the photoproduction origin of the observed J/$\psi$
excess. The quantitative understanding of this low-$p_T$ excess poses
significant theoretical challenges since the J/$\psi$ photoproduction depends
on the collision dynamics as well as on the photon-flux and the photonuclear
cross section. In this proceeding, we present the latest ALICE measurements on
J/$\psi$ photoproduction cross section in peripheral Pb-Pb collisions, with
emphasis on the new forward measurement in the dimuon decay channel at
$\sqrt{s_{\rm NN}}$ = 5.02~TeV. These results will be discussed and compared to
several model calculations of J/$\psi$ photoproduction in Pb-Pb collisions with
nuclear overlap.
