A stratified variety admits a Bruhat (resp., Kazhdan-Lusztig) atlas if it can be covered by open charts isomorphic to opposite Bruhat cells (resp., Kazhdan-Lusztig varieties) in some Kac-Moody flag manifold via stratified isomorphisms. In the first part of this thesis, we construct an anticanonical stratification on the wonderful compactification of the symmetric space $PSO(2n)/SO(2n-1)$ and show that the open charts are stratified-isomorphic to certain (opposite) Bruhat cells in the type $D_{n+1}$ flag manifold. In the second part of this thesis, we show that the partial flag manifold $G/P$ with the projected Richardson stratification has a Kazhdan-Lusztig atlas, with each chart stratified-isomorphic to a Kazhdan-Lusztig variety in the affine flag manifold of the loop group $\widehat{G}$ of $G$. Furthermore, we give an affine analogue of Fulton's matrix Schubert varieties in the appendix and use it as a computational tool to illustrate an example of a Kazhdan-Lusztig atlas on a partial flag variety in type A.