Hessenberg varieties are subvarieties of a flag variety whose topology are related to other research areas such as hyperplane arrangements and graph theory. In this talk, we show that the set of certain monomials form a basis for the cohomology ring of a regular nilpotent Hessenberg variety. The monomial basis is important in Schubert calculus in the case of the flag variety. This is joint work with Megumi Harada, Satoshi Murai, Martha Precup, and Julianna Tymoczko.