Abstract: I will consider the general setting of an algebraic group acting with finitely many orbits on the flag variety, and describe several problems related to the geometry of the orbits and their closures in this setting. Examples include parametrizing the orbits; understanding certain partial orders on the orbit sets; computing cohomological invariants for orbit closures; determining when an orbit has smooth closure; and computing its singular locus if it does not. In the study of such problems, algebraic combinatorics inevitably enters the picture. I will focus mostly on the cases which are currently most well-understood: Schubert varieties and symmetric varieties.