Here is a MO post asking a question that I’ve had in mind for a while: Why higher category theory?
Having studied some elementary topos before, I have been interested in higher topos since I attended a summer school lecture last year. Besides formal generalisations, I expect to see applications which provide new results or meaningful insights. Though the meaning of “new” and “useful” very much diverse between different mathematical cultures.
Here are some important applications of higher categories in K-theory (added on Mar 05, 2020), suggested by my supervisor Schlichting.