  We consider two nested billiards in $\mathbb R^d$, $d\geq3$, with
$C^2$-smooth strictly convex boundaries. We prove that if the corresponding
actions by reflections on the space of oriented lines commute, then the
billiards are confocal ellipsoids. This together with the previous analogous
result of the author in two dimensions solves completely the Commuting Billiard
Conjecture due to Sergei Tabachnikov. The main result is deduced from the
classical theorem due to Marcel Berger saying that in higher dimensions only
quadrics may have caustics. We also prove versions of Berger's theorem and the
main result for billiards in spaces of constant curvature: space forms.
