  Systematic studies of charge-dependent two- and three-particle correlations
in Pb-Pb collisions at $\sqrt{\it{s}_\mathrm{{NN}}} = $ 2.76 and 5.02 TeV used
to probe the Chiral Magnetic Effect (CME) are presented. These measurements are
performed for charged particles in the pseudorapidity ($\eta$) and transverse
momentum ($p_{\rm{T}}$) ranges $\left|\eta \right| < 0.8$ and $0.2 <
p_{\mathrm{T}} < 5$ GeV/$c$. A significant charge-dependent signal that becomes
more pronounced for peripheral collisions is reported for the CME-sensitive
correlators $\gamma_{1,1} = \langle \cos (\varphi_{\alpha} +\varphi_{\beta} -
2\Psi_{2}) \rangle$ and $\gamma_{1,-3} = \langle \cos (\varphi_{\alpha}
-3\varphi_{\beta} + 2\Psi_{2}) \rangle$. The results are used to estimate the
contribution of background effects, associated with local charge conservation
coupled to anisotropic flow modulations, to measurements of the CME. A
blast-wave parametrisation that incorporates local charge conservation tuned to
reproduce the centrality dependent background effects is not able to fully
describe the measured $\gamma_{1,1}$. Finally, the charge and centrality
dependence of mixed-harmonics three-particle correlations, of the form
$\gamma_{1,2} = \langle \cos (\varphi_{\alpha} +2\varphi_{\beta} - 3\Psi_{3})
\rangle$, which are insensitive to the CME signal, verify again that background
contributions dominate the measurement of $\gamma_{1,1}$.
