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.
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