  In chemistry, a catalyst is a substance which enables a chemical reaction or
increases its rate, while remaining unchanged in the process. Instead of
chemical reactions, quantum catalysis enhances our ability to convert quantum
states into each other under physical constraints. The nature of the
constraints depends on the problem under study and can arise, e.g., from energy
preservation. This article reviews the most recent developments in quantum
catalysis and gives a historical overview of this research direction. We focus
on the catalysis of quantum entanglement and coherence, and also discuss this
phenomenon in quantum thermodynamics and general quantum resource theories. We
review applications of quantum catalysis and also discuss the recent efforts on
universal catalysis, where the quantum state of the catalyst does not depend on
the states to be transformed. Catalytic embezzling is also considered, a
phenomenon that occurs if the catalyst's state can change in the transition.
