A complex airplane is defined by the United States Federal Aviation Administration as an aircraft that has all of:
In the U.S., students generally train for their first pilot certificate in an aircraft with fixed landing gear, a fixed-pitch propeller, and movable flaps.
Before or after earning the private pilot certificate (PPL) (usually after), a pilot can be trained in complex aircraft operation by a flight instructor. When the pilot has demonstrated proficiency in complex aircraft, the flight instructor endorses the pilot's logbook and the pilot is said to have a "complex endorsement".
The FAA requires a pilot to have experience in, and take the practical test in, a complex airplane in order to earn the commercial pilot certificate (CPL) and the flight instructor certificate (CFI).