Motivated by results of Knutson-Lam-Speyer and Snider, Bruhat atlases were introduced by He, Knutson
and Lu. A Bruhat atlas on a stratified manifold consists of charts $U$ with stratified isomorphisms
$ U\cong X^w_o $, where $X^w_o$ is a Bruhat cell in some Kac-Moody flag variety. Interesting examples of manifolods with Bruhat atlases include partial flag manifolds and the wonderful compactifications of groups. This remarkable structure is very restrictive, so one hopes to obtain a complete classification of manifolds with Bruhat
atlases. We will discuss the classification problem; focusing primarily on smooth toric surfaces.