ABSTRACT. We introduce refutationally complete superposition calculi for intentional and extensional lambda-free higher-order logic, a formalism that allows partial application and applied variables. The intentional variants perfectly coincide with standard superposition on first-order clauses. The calculi are parameterized by a well-founded term order that need not be compatible with arguments, making it possible to employ the lambda-free higher-order lexicographic path and Knuth-Bendix orders. We implemented the calculi in the Zipperposition prover and evaluated them on TPTP benchmarks. They appear promising as a stepping stone towards complete, efficient automatic theorem provers for full higher-order logic.