  The ALICE detector provides excellent capabilities to study quarkonium
production at the Large Hadron Collider (LHC). Heavy quarkonia, bound states of
charm or beauty quark anti-quark pairs such as the $\mathrm{J/\psi}$, are
expected to be produced by initial hard processes. Thus they will provide
insight into the earliest and hottest stages of AA collisions where the
formation of a Quark-Gluon Plasma (QGP) is expected. Furthermore,
high-precision data from pp collisions represent an essential baseline for the
measurement of nuclear modifications in heavy-ions and serve also as a crucial
test for several models of quarkonium hadroproduction. In addition, the study
of pA collisions allows to investigate nuclear modifications due to Cold
Nuclear Matter (CNM) effects. In ALICE, $\mathrm{J/\psi}$ were measured in pp
and Pb--Pb collisions down to $p_{\mathrm T}$ = 0 via their di-electron decay
channel in the central barrel ($|y| <$ 0.8). Results on the nuclear
modification factor ($R_{\mathrm{AA}}$) at central rapidities in Pb--Pb
collisions at $\sqrt{s_{\mathrm{NN}}} = 2.76$ TeV will be shown and their
implications discussed. A separation of the prompt and non-prompt components is
also possible down to $p_{\mathrm T}$ of the $\mathrm{J/\psi}$ of 2 GeV/$c$.
