  A Riemannian manifold is said to be almost positively curved if the sets of
points for which all $2$-planes have positive sectional curvature is open and
dense. We show that the Grassmannian of oriented $2$-planes in $\mathbb{R}^7$
admits a metric of almost positive curvature, giving the first example of an
almost positively curved metric on an irreducible compact symmetric space of
rank greater than $1$. The construction and verification rely on the Lie group
$\mathbf{G}_2$ and the octonions, so do not obviously generalize to any other
Grassmannians.
