The FAI or
Federation Aeronautique Internationale is the international organisation based
in Paris, France, which governs all types of aviation related sporting
disciplines throughout the world. The FAI is the body behind the world aerobatic
championships.
Aerobatics competitions in the U.S. are organised by the International Aerobatic Club that was formed in 1970 and in 1981 after a lot of internal squabbling with another organisation was recognised as the U.S. national body. There are several dozen regional competitions each year in the US and one big one every two years.
The aerobatic
organisation in Australia is the Australian Aerobatic Club or AAC.