We prove that a hyperplane in a CAT(0) cubical complex X has no
self-intersections and separates X into two convex complementary components.
These facts were originally proved by Sageev. Our argument shows that his
theorem is a corollary of Gromov's link condition.

We also give new arguments establishing some combinatorial properties of
hyperplanes. We show that these properties are sufficient to prove that the
0-skeleton of any CAT(0) cubical complex is a discrete median algebra, a fact
that was first proved by Roller.