What properties do all waves share, and how does the electromagnetic spectrum connect visible light to radio waves and X-rays?
Waves transfer energy from place to place without transferring matter. The medium (if there is one) oscillates but doesn't travel with the wave. Transverse waves (light, EM radiation, water waves) oscillate perpendicular to the direction of travel; longitudinal waves (sound, seismic P-waves) oscillate parallel to it. Every wave has an amplitude, frequency, wavelength, and wave speed — related by the wave equation v = fλ.
The electromagnetic (EM) spectrum is a family of transverse waves, all travelling at 3 × 10⁸ m/s in a vacuum. They differ only in frequency and wavelength. The entire spectrum — from radio waves to gamma rays — carries different energy per photon (E = hf), which determines both their uses and their hazards. Infrared is heat radiation; ultraviolet causes sunburn; X-rays and gamma rays are ionising and damage DNA.