An early Adelaide airport was built in 1921, allowing a mail service between Adelaide and Sydney. The aerodrome was built on 24 ha (59 acres) of land in Albert Park, now part of the neighbouring suburb of Hendon, which took over from the Northfield Aerodrome. The demand on aviation soon grew substantially, with Parafield Airport being developed in 1927 to enable the first passenger airline services from Adelaide. With a further growth in aviation, a new site for the current Adelaide Airport was selected in the suburb of West Beach in January 1946, which was then split to form the dedicated Adelaide Airport suburb in 1991. An alternative airport site was also considered in Port Adelaide, which included a seaplane facility, but was considered inferior and too far from the central business district at roughly 14 kilometres (8.7 miles) away.

Construction in the chosen site took place in 1947 and flights commenced in 1954, with Parafield Airport being turned into a private and military aviation facility, now operating as a public training airport. With regular transport operations commencing on 16 February 1955, an annexe to one of the large hangars at the airport served as the first passenger terminal, until the Federal Government provided funds for the construction of a temporary building.

In 1982, a contract was awarded to Baulderstone to prepare the airport for wide bodied aircraft service, jets in particular being the Airbus A300 and Boeing 767 aircraft which would open the door to long-haul services.

In May 1998, long-term leases of Adelaide Airport and Parafield Airport were sold by Federal Airports Corporation to Adelaide Airport Limited, a consortium of Manchester Airport, Serco, UniSuper and Macquarie Bank. As at December 2023, the shareholders of Adelaide Airport Limited were UniSuper (51%), Hostplus (15%), IFM Investors (15%), Igneo Infrastructure Partners (15%) and Perron Group (4%).

In July 1998, the runway was extended by 570 metres to 3.1 kilometres. Prior to the 2005 terminal redevelopment, the original separate terminals had only two aircraft bays and a single jetbridge with limited space for passengers, check-in desks were also small and waiting space was limited. A new control tower was also built west of the terminal, with the old control tower maintained for additional operations.

In October 2006, the new terminal was named the Capital City Airport of the Year at the Australian Aviation Industry Awards in Cairns. In March 2007, Adelaide Airport was rated the world's second-best airport in the 5–15 million passengers category at the Airports Council International 2006 awards.

Following the successful terminal redevelopment, plans were announced for another expansion of the terminal in July 2007, including the addition of more aerobridges as well as demolition of the old international terminal building. However, the demolition was delayed until November 2016 and was finally concluded in April 2018, this would make way for expanded landside facilities as well as further future expansions of the main terminal.

On 5 August 2008 Tigerair Australia confirmed that Adelaide Airport would become the airline's second hub which would base two of the airline's Airbus A320s by early 2009, this was followed by a third A320 in early 2010. In 2011, Tigerair announced it would shut down its operations from Adelaide and Avalon following the airlines 2011 suspension and reduction in flights.

In 2011, Adelaide Airport encountered major problems during the eruption of Puyehue volcano in Chile, with the ash cloud caused by the volcano resulting in flight cancellations nationwide, this left over 40,000 passengers stranded in Adelaide.

On 11 October 2022, it was discovered that at around 10am local time, security screening equipment had failed half an hour earlier, leading to the evacuation of the terminal and re-screening of approximately 2,000 passengers.

On 2 September 2025, Jetstar based another Airbus A320ceo at Adelaide Airport to support an increase in services, boosting the total number of aircraft based at the airport to six.