  Let $B$ be a Borel subgroup of $\mathrm{GL}_n(\mathbb{C})$ and $\mathbb{T}$ a
maximal torus contained in $B$. Then $\mathbb{T}$ acts on
$\mathrm{GL}_{n}(\mathbb{C})/B$ and every Schubert variety is
$\mathbb{T}$-invariant. We say that a Schubert variety is of complexity $k$ if
a maximal $\mathbb{T}$-orbit in $X_w$ has codimension $k$. In this paper, we
discuss topology, geometry, and combinatorics related to Schubert varieties of
complexity one.
