Aeroplanes were a form of transport that travelled in the air. Whilst in flight, passengers were cared for by air stewardesses. (TV: Logopolis)
The Fifth Doctor "expected" to be able to fly a plane, and would have managed to do so properly had the Daleks not attacked the craft. (AUDIO: Fugitives) The Eleventh Doctor was due for a lesson in flying a bi-plane in 1911. However, it may have in fact been a knitting lesson. (TV: The Impossible Astronaut) He later claimed that he couldn't fly a plane at all. (TV: The Bells of Saint John)
History
Leonardo da Vinci tried to build a flying machine ("a sort of aeroplane", as Steven Taylor put it) after the Monk discussed with him the principles of powered flight. (TV: "Checkmate")
Aeroplans were finally invented by the Wright Brothers in the early 20th century. Lethbridge-Stewart's father could remember reading about their first flight. (PROSE: The Dying Days) By 1904, the brothers had "barely flown sixty feet yet". (PROSE: Gudok)
In 1932, Japanese aircraft launched from the aircraft carrier Hosho attacked Shanghai. In 1937, they used Mitsubishi Ki-15s and Mitsubishi A5Ms on air raids against China. (PROSE: The Shadow of Weng-Chiang)
In 1941, Edwin Bracewell used gravity bubbles on modified Spitfires, sending them into space. There they disabled the Dalek flying saucer's transmitter. (TV: Victory of the Daleks)
In 1964, a light aircraft nearly collided with the TARDIS, which had materialised in mid-air over Alaska. (AUDIO: The Land of the Dead)
The same year, a plane which left New York City with a crew of eleven and hundreds of passengers landed in London with no-one aboard, not even the pilot. The matter was investigated by Professor Rachel Jensen of the Intrusion Countermeasures Group. (AUDIO: State of Emergency)
In 1966, Chameleon Tours operated select flights from Gatwick Airport. They kidnapped passengers, taking them to a space station, and putting them in suspended animation whilst miniaturised. They were then replaced by Chameleons. (TV: The Faceless Ones)
In the 1970s, UNIT deployed an RAF air strike that destroyed the Krynoid at Chase Mansion. (TV: The Seeds of Doom)
In 1982, two Concordes, Golf Victor Foxtrot and Golf Alpha Charlie, were taken 140 million years into the past when they flew into a time corridor. (TV: Time-Flight)
In 2007, the Sky Gypsy came through the Cardiff Space-Time Rift. It carried Diane Holmes, John Ellis, and Emma-Louise Cowell. Diane attempted to return through it by flying again in the same conditions. (TV: Out of Time)
In 2013, Miss Kizlet, who had control of all of humanity through the Wi-Fi, made everyone on an aeroplane, including the pilot, fall asleep, and the plane came crashing down towards George Maitland's house. Using the TARDIS, the Eleventh Doctor and Clara Oswald entered the plane and changed its course to prevent a crash. Everyone on the airplane woke up when the Doctor disabled the plane's Wi-Fi. (TV: The Bells of Saint John)
Missy froze all the Earth's planes, totaling 4,165 at the time, in the world to get Clara's attention. She later released them. (TV: The Magician's Apprentice)
Other Realities
Dream World
In a dream future created by the Kantrofarri, Clara learned to fly a plane. (TV: Last Christmas)