  Grassmann manifolds $G_{k,n}$ are among the central objects in geometry and
topology. The Borel picture of the mod 2 cohomology of $G_{k,n}$ is given as a
polynomial algebra modulo a certain ideal $I_{k,n}$. The purpose of this paper
is to understand this cohomology via Gr\"obner bases. Reduced Gr\"obner bases
for the ideals $I_{k,n}$ are determined. An application of these bases is given
by proving an immersion theorem for Grassmann manifolds $G_{5,n}$, which
establishes new immersions for an infinite family of these manifolds.
