  We show that the maximal number of planes in a complex smooth cubic fourfold
in ${\mathbb P}^5$ is $405$, realized by the Fermat cubic only; the maximal
number of real planes in a real smooth cubic fourfold is $357$, realized by the
so-called Clebsch--Segre cubic. Altogether, there are but three (up to
projective equivalence) cubics with more than $350$ planes.
