The chromatic quasisymmetric function of a labeled graph was introduced by Shareshian and myself as a refinement of Stanley’s chromatic symmetric function. We conjectured a refinement of the long standing Stanley- Stembridge e-positivity conjecture, and formulated an algebro-geometric approach to proving this refined conjecture involving Hessenberg varieties. Significant progress in this direction has recently been made by Brosnan and Chow and by Guay-Paquet. In this talk, I will discuss the connection with Hessenberg varieties, and also present some new directions, including results on generalizations to directed graphs obtained by my student Brittney Ellzey