We will discuss a class of topological G-spaces generalizing the classical flag manifolds F(G,T)= G/T of compact connected Lie groups. These spaces — which we call the m-quasi-flag manifolds F_m(G,T) --- are natural `homotopical approximations' of F(G,T). Many cohomological properties and structures related to F(G,T) extend naturally to F_m(G,T) (some of these, e.g. equivariant K-theory, will be discussed in the talk). The construction of quasi-flag manifolds uses tools from three areas of topology: the theory of homotopy decompositions of classifying spaces, the Goresky-Kottwitz-MacPherson (GKM) theory of moment graphs, and the theory of co-affine stacks (a very convenient geometric approach to rational homotopy theory proposed by B. Toen (2006) and developed by J. Lurie (2011) in the framework of derived algebraic geometry). We will try to provide a gentle introduction to all three areas.