Airplanes have been flown by remote-control since 1917. As Wikipedia notes:

In 1917, Archibald Low as head of the RFC [Britain's Royal Flying Corps] Experimental Works, was the first person to use radio control successfully on an aircraft. *** There were also [during the 1930s] remotely controlled cutters and experimental remotely controlled planes in the Red Army. In the 1930s Britain developed the radio controlled Queen Bee, a remotely controlled unmanned Tiger Moth aircraft for a fleet’s gunnery firing practice. The Queen Bee was superseded by the similarly named Queen Wasp, a later, purpose built, target aircraft of higher performance.

As the Norfolk and Suffolk Aviation Museum notes, President John F. Kennedy’s older brother flew a secret mission involving the remote-control flying of a bomb-laden airplane to attack Nazi targets inside France:

On the 31st July 1944 a U.S.N. special air unit, codenamed Project Anvil, moved to Fersfield from Dunkeswell in Devon. The mission was to involve the use of explosive-laden PB4Y-1 Liberator bombers under radio control. The crew of two, Lt Joe Kennedy (pilot), and Lt. Wilford John Willy (radio control technician/co-pilot), were to take off with 21,150 lbs of Torpex in 347 boxes and establish radio control of the Liberator by a Ventura mother-ship. Once full control was established and tested, at a pre-determined point the crew would parachute from the aircraft through the nose wheel bay emergency exit and the bomber would continue the rest of its mission under radio control, finally crashing onto the target.

In addition, Norad has been able to fly planes remotely for many decades:

NORAD (the North American Air Defense Command) had at its disposal a number of U.S. Air Force General Dynamics F-106 Delta Dart fighter aircraft configured to be remotely flown into combat as early as 1959 under the auspices of a program know as SAGE. These aircraft could be started, taxied, taken off, flown into combat, fight, and return to a landing entirely by remote control, with the only human intervention needed being to fuel and re-arm them.

As Wikipedia explains:

The Semi-Automatic Ground Environment (SAGE) was an automated control system for tracking and intercepting enemy bomber aircraft used by NORAD from the late 1950s into the 1980s. In later versions, the system could automatically direct aircraft to an interception by sending instructions directly to the aircraft’s autopilot. *** In normal operation, communications between the SAGE centers and the interceptor aircraft was relayed via radio equipment at the radar sites, which were more widely spread out than the SAGE centers themselves. A properly equipped aircraft, like the F-106 Delta Dart, could feed the SAGE directions into the autopilot and fly “hands off” to the interception.

NASA and the FAA flew a plane by remote control in 1984:

In 1984 NASA Dryden Flight Research Center and the Federal Aviation Administration (FAA) teamed-up in a unique flight experiment called the Controlled Impact Demonstration (CID), to test the impact of a Boeing 720 aircraft using standard fuel with an additive designed to suppress fire. *** On the morning of December 1, 1984, a remotely controlled Boeing 720 transport took off from Edwards Air Force Base (Edwards, California), made a left-hand departure and climbed to an altitude of 2300 feet. It then began a descent-to-landing to a specially prepared runway on the east side of Rogers Dry Lake. Final approach was along the roughly 3.8-degree glide slope.

Indeed, prior to 9/11, remote-controlled planes could fly up to 8,600 miles (from the April 24, 2001 edition of Britain’s International Television News).

One day after 9/11, an article appeared on the top science and technology news service stating “hijackings could be halted in progress with existing technologies, say aviation researchers”. The article quoted a transportation expert as saying:

“Most modern aircraft have some form of autopilot that could be re-programmed to ignore commands from a hijacker and instead take direction from the ground . . . .”

See also this article, in which the former head of British Airways “suggested . . . that aircraft could be commandeered from the ground and controlled remotely in the event of a hijack.”

Some have speculated that remote control played a part in 9/11:

And some allege that the use of remote control could explain some of the strange behavior by the 9/11 planes.

Indeed, more than 40 years ago, the U.S. Joint Chiefs of Staff suggested shooting down a military drone airplane, pretending it was a real airplane, and then blaming the attack on the Cubans as a way to justify an invasion of Cuba. See the following ABC news report; the official documents; and watch this interview with the former Washington Investigative Producer for ABC’s World News Tonight with Peter Jennings.

Interestingly, NORAD - which is the military air defense agency responsible for protecting the U.S. mainland - had run drills for several years of planes being used as weapons against the World Trade Center and other U.S. high-profile buildings, and "numerous types of civilian and military aircraft were used as mock hijacked aircraft". And coincidentally, Fox TV aired a fictional drama 6 months before 9/11, in which the U.S. government intended to fly a plane into the World Trade Center via remote control and blame it on terrorists.

Note: While some claim that remote control played a part in 9/11, a separate - but interesting - question, is whether remote control could and should have been used to safely land the hijacked airplanes. Given that Al Qaeda flying planes into the World Trade Center and Pentagon was wholly foreseeable, and hijackings could be stopped using existing equipment, why wasn't the equipment used to stop this type of attack? In other words, why didn't ground control have the ability to override the hijacked airlines to safely land them and take control of the aircraft?