Also: Leerflug, Empty sector, Non-revenue flight
In business aviation, empty flights are routine. A private jet flies clients from Hamburg to Nice, for example, and then has to return to base or continue to its next job. That transfer without passengers is the empty flight.
For operators, empty flights are a cost: fuel, crew and fees still apply. That is why many of them are marketed as empty legs, so that at least part of the cost is recovered.
In everyday use, the German term Leerflug and empty leg mean much the same thing. Strictly speaking, the empty flight is the operational event and the empty leg is the bookable offer that comes from it.