  This is a survey on two closely related subjects. First, we review the study
of topological structure of `finite type' components of spaces of Bridgeland's
stability conditions on triangulated categories. The key is to understand
Happel-Reiten-Smalo tilting as tiling of cells. Second, we review topological
realizations of various Fukaya type categories, namely cluster/Calabi-Yau and
derived categories from surfaces. The corresponding spaces of stability
conditions are of `tame' nature and can be realized as moduli spaces of
quadratic differentials due to Bridgeland-Smith and
Haiden-Katzarkov-Kontsevich.
