Airponics (or aeroponics) is the process of cultivating plants without soil. This is achieved by suspending the plants in an oxygen-rich environment and then misting their roots with a nutrient solution.
In late 2265, Lieutenant Commander Spock was performing a series of aeroponics experiments aboard the USS Enterprise. Following the ship's departure from Starbase 33, Spock asked Captain James T. Kirk for permission to leave the bridge so that he could check on them. (TOS - My Brother's Keeper novel: Constitution)
After the USS Voyager was catapulted into the Delta Quadrant by the Caretaker in 2371 and being stranded over 70,000 light-years from Earth it became necessary to find a means of growing their own food supplies as replicator power wouldn't always be guaranteed. At Kes's suggestion, Captain Kathryn Janeway approved the establishment of an airponics garden and hydroponics bay aboard Voyager which would be overseen by Kes until her departure in 2374. (VOY episode: "Parallax")