  After a review over past experiments and theoretical requirements from
low-energy flavour physics, we argue that the possibility of low-mass
leptoquarks (LQ) cannot be fully excluded due to the assumptions made in the
measurements. Therefore we propose to search for the pair production of
low-mass leptoquarks in Pb-Pb ultra-peripheral collisions with the least model
dependence (assuming a small coupling constant $\lesssim0.1$). There are a
couple of advantages: 1) high photon flux provides high production rate for low
mass LQs; 2) the background contamination is much lower than that in $p$-$p$
collisions. The analysis strategy permits a leptoquark to decay to all possible
lepton-plus-quark modes. Taking the scalar LQ, $S_3$, with an electric charge
$|q|=\frac{4}{3}e$, as example, the mass point of 100 GeV can be excluded at
the 95~\% confidence level using a dataset of 4~pb$^{-1}$ Pb-Pb
ultra-peripheral collisions at $\sqrt{s}=5.02$ TeV and the performance of the
ATLAS detector in Run~2. The proposed method also applies to searching for
high-mass LQs in $p$-$p$ collisions as long as the LQ pair production mechanism
dominates.
