Schubert varieties and Hessenberg varieties are subvarieties of the flag variety with connections to both algebraic combinatorics and representation theory. I will discuss joint work with Martha Precup and John Shareshian in which we investigate which Schubert varieties are Hessenberg varieties. In the process, we give a restriction on the Euler characteristic of codimension-one Hessenberg varieties, as well as a description of the singular loci of regular codimension-one Hessenberg varieties.