  Hessenberg varieties $\mathcal{H}(X,H)$ form a class of subvarieties of the
flag variety $G/B$, parameterized by an operator $X$ and certain subspaces $H$
of the Lie algebra of $G$. We identify several families of Hessenberg varieties
in type $A_{n-1}$ that are $T$-stable subvarieties of $G/B$, as well as
families that are invariant under a subtorus $K$ of $T$. In particular, these
varieties are candidates for the use of equivariant methods to study their
geometry. Indeed, we are able to show that some of these varieties are unions
of Schubert varieties, while others cannot be such unions.
  Among the $T$-stable Hessenberg varieties, we identify several that are {\it
GKM spaces}, meaning $T$ acts with isolated fixed points and a finite number of
one-dimensional orbits, though we also show that not all Hessenberg varieties
with torus actions and finitely many fixed points are GKM.
  We conclude with a series of open questions about Hessenberg varieties, both
in type $A_{n-1}$ and in general Lie type.
