Leo-III is an automated theorem prover for extensional type theory with Henkin semantics and choice. Reasoning with primitive equality is enabled by adapting paramodulation-based proof search to higher-order logic. The prover may cooperate with multiple external specialist reasoning systems such as first-order provers and SMT solvers. Leo-III is compatible with the TPTP/TSTP framework for input formats, reporting results and proofs, and standardized communication between reasoning systems, enabling, e.g., proof reconstruction from within proof assistants such as Isabelle/HOL. Leo-III supports reasoning in polymorphic first-order and higher-order logic, in many quantified normal modal logics, as well as in different deontic logics. Its development had initiated the ongoing extension of the TPTP infrastructure to reasoning within non-classical logics.