  The Jahn-Teller effect is one of the most fundamental phenomena important not
only for physics, but also for chemistry and material science. Solving the
Jahn-Teller problem and taking into account strong electron correlations we
show that quantum entanglement of the spin and orbital degrees of freedom via
spin--orbit coupling strongly affects this effect. Depending on the number of
$d$ electrons it may quench (electronic configurations $t_{2g}^2$, $t_{2g}^4$,
and $t_{2g}^5$), partially suppress ($t_{2g}^1$) or in contrast induce
($t_{2g}^3$) Jahn-Teller distortions. Moreover, in certain situations,
interplay between the Jahn-Teller effect and spin--orbit coupling promotes
formation of the "Mexican hat" energy surface facilitating various quantum
phenomena.
