Multiple Flag Varieties

This paper is a review of results on multiple flag varieties, i.e., varieties of the form $G/P_1\times\dots\times G/P_r$. We provide a classification of multiple flag varieties of complexity $0$ and $1$ and results on the combinatorics and geometry of $B$-orbits and their closures in double cominuscule flag varieties. We also discuss questions of finiteness for the number of $G$-orbits and existence of an open $G$-orbits on a multiple flag variety.