Open problems in one-parameter operator semigroups theory

One-parameter strongly continuous semigroups of linear bounded operators on Banach spaces (also known as $C_0$-semigroups) are a fundamental operator-theoretic tool used in the study of linear and non-linear evolution PDEs arising in physics, probability, control theory and other areas of science and technology, including quantum theory, transportation problems and finance. Since 2021 the annual online conference on one-parameter semigroups of operators (OPSO) offers an opportunity for researchers worldwide to discuss the current state of the art and exchange knowledge. On every OPSO conference open problems are proposed, collected and discussed, some of them are found to be solved and hence get crossed out from the list. In this paper we provide a collection of open problems of semigroup theory that were proposed by participants of OPSO 2021--2024 conferences and have not been solved, yet. For each problem some comments on the relevance and and history of the problem are provided.