Hessenberg varieties are subvarieties of the full flag variety, and they form a relatively new research subject which was introduced by De Mari-Procesi-Shayman around 1990. Particular examples are the flag variety itself, Springer fibres, the Peterson variety, and the permutohedral variety. Similarly to Schubert varieties, it has been found that geometry, combinatorics, and representation theory interact nicely on Hessenberg varieties, and there are still many things to be studied.

In this talk, I will give a survey of recent developments on Hessenberg varieties, especially from algebro-geometric and combinatorial points of view. The goal of this talk is to advertise that Hessenberg varieties can be studied from various perspectives.

(Keywords: representations of symmetric groups, hyperplane arrangements, toric degenerations, integrable systems)