What is "the hunting of wild animals in their natural state by means of a trained bird of prey" called?

I believe that sport hunting is barbaric and cruel to prey. But as a hobby because you love your falcon and train it, if the prey is consumed or used, or not wanted such as rats or snakes, then I might participate. Yet, probably my POV is let the falcon live wild and free.

I've not heard of falconaey

San, agreed i have no problem with hunting for food but to do it for trophies or "the thrill" is disgusting

I love birds 😉😉