In this video of AC/DC - The Jack, Angus Young can be seen running around for long distances while playing his electric guitar.
It's pretty obvious there's no cable. So how does the signal get to the amplifier?
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Wireless instrument systems are becoming more common among professional guitarist as the technology improves and prices come down. Using a wireless transmitter frees up the guitarist to move around on stage without worry about tripping (or his band mates tripping) over the instrument cables. A wireless guitar system is similar to a wireless mic. There is a body pack transmitter (which many guitarist attach to their guitar strap) with a short 1/4" cable that you plug into your guitar as you would a guitar cable. This transmitter sends the signal to a receiver plugged in to the amp or PA system. They use a specific radio frequency band similar to a wireless microphone. Many musicians are also using wireless "in-ear" monitors (known as "ears") on stage as well. Then while they are running around on stage and out in the audience, they hear a consistent monitor mix.
Angus young was one of the earliest users of wireless guitar technology. Specifically he used the Schaffer-Vega Diversity System, which was one of the first stable wireless systems. Some AC/DC fans credit this system, which allowed the user to boost the volume of the system, with creating Angus's most recognised live and studio guitar tones. As such replicas are now being manufactured.
I've heard of some guitarists having a wireless system built into their guitars by a tech or luthier. The artist Prince springs to mind, as he often appeared to be playing an unplugged guitar, without even a visible wireless system connected to the output jack. The guitar manufacturer 'Micro Frets' introduced a built-in wireless system as early as 1969, using a suitably futuristic looking system built into the upper horn of their creations.
More prosaically, some guitarists are simply unplugged. The British TV show 'Top of the Pops' featured almost exclusively mimed performances, and they never usually went to the bother of running cables out to the guitars on stage.