I will introduce Hessenberg varieties, an important class of subvarieties of the flag variety shown to govern certain representation theoretic and combinatorial phenomena related to symmetric functions and generalizations. The geometry of these varieties remains somewhat mysterious: they support $\mathbb C^*$ actions, though they are not typically GKM spaces. They have a paving by affine cells, yet their cohomology rings do not always satisfy Poincare’ duality. In some circumstances, they are known to be fibers of a flat family of Hessenberg varieties. In this talk, I will describe joint work with Martha Precup proving that semisimple Hessenberg varieties over a line in the minimal sheet of the Lie algebra are in a flat family with special fiber given by a nilpotent Hessenberg variety. We will also describe some cohomological consequences of the result.