Hawk Moth's lair is a room or observatory at an unknown location. Here, Hawk Moth stores many white butterflies that, once corrupted, become akumas, which he uses to akumatize his victims.
The room is large, dome-like and dark when Hawk Moth is not using an akuma. It has a large window with a butterfly (or flower) design that opens whenever Hawk Moth plans to send an akuma to a person who feels strong negative emotions. In the far back of the room, an octagonal shape is on the ground, which might be an exit/entrance to the lair.
Once a villain gets defeated, the window closes, encasing the room in darkness once again.