Moduli spaces for Kuznetsov components of Fano varieties, and applications

I will give a survey of recent results on moduli spaces of stable objects in Kuznetsov components of various Fano varieties. I will focus on concrete geometric applications; this includes, for example, Torelli-type statements for Fano threefolds, and on the geometry of Hassett-special cubic fourfolds via Brill-Noether loci inside such moduli spaces.