Generalized descriptive set theory has mostly been developed for uncountable cardinals satisfying the condition κ<κ=κ (thus in particular for κ regular). More recently the case of uncountable cardinals of countable cofinality has attracted some attention, partially because of its connections with very large cardinal axioms like I0. In this talk I will survey these recent developments and propose a unified approach which potentially could encompass all possible scenarios (including singular cardinals of arbitrary cofinality).