Completeness for modal predicate logics

In modal predicate logic the problem of semanticaI completeness is very nontrivial. Kripke semantics is too weak, and it is not clear how to improve it. In this talk we give a brief overview of the field and present proofs of some recent results on completeness and incompleteness. An earlier version of the talk was given at Workshop on first-order modal and temporal logics (Ljubljana 2023).