  The $B^0_{s,d} \to J/\psi \eta^{(\prime)}$ decays provide new terrain for
exploring CP violation. After briefly discussing $\eta$-$\eta'$ mixing, we
analyse the effective lifetimes and CP-violating observables of the $B_s$
channels, which allow us to probe New-Physics effects in $B^0_s$-$\bar B^0_s$
mixing. We have a critical look at these observables and show how hadronic
corrections can be controlled by means of the $B_d$ decays. Using measurements
of the $B^0_{s,d}\to J/\psi \eta^{(\prime)}$ branching ratios by the Belle
collaboration, we discuss tests of the $SU(3)_F$ flavour symmetry of strong
interactions, obtain the first constraints on the hadronic parameters entering
the $B^0_{s,d} \to J/\psi \eta$ system, and predict the $B^0_d\to J/\psi \eta'$
branching ratio at the $5\times10^{-6}$ level. Furthermore, we present
strategies for the determination of the $\eta$-$\eta'$ mixing parameters from
the $B^0_{s,d} \to J/\psi \eta^{(\prime)}$ observables. We also observe that
the $B^0_{s,d} \to J/\psi \eta$ and $B^0_{s,d} \to J/\psi \eta'$ decays are -
from a formal point of view - analogous to the quark-antiquark and tetraquark
descriptions of the $f_0(980)$ in the $B^0_{s,d} \to J/\psi f_0(980)$ channels,
respectively.
