Cavity resonators have a multitude of applications in radio-frequency, microwave, and vacuum electronics. Cavity resonators are used as frequency selective filters and oscillators. In vacuum electronic devices and charged particle accelerators cavities are used to couple energy into and out of charged particle beams. Typically cavity resonators are optimized for single mode operation and are limited to sinusoidal waveforms and interactions. Harmonic cavities have many of the same applications, but because they resonate many axially symmetric harmonic modes simultaneously, the superposition of these modes is an on-axis arbitrary waveform. Harmonic cavities have both passive and active applications. When a periodic charged particle beam passes through a harmonic cavity whose resonances are at the same frequency as the beam's periodicity, the beam induces these cavity modes to resonate. The superposition of these modes, on axis, has a voltage vs. time waveform that ideally mirrors the current vs. time of the beam that induced it. Harmonic cavities can also be used to apply arbitrary waveforms to charged particle beams when driven externally. These waveforms can be used to modulate, shape, and accelerate the beam. A harmonic cavity whose axis is tilted to the beamline can be used to change the trajectory of selected particle bunches by delivering them fast kicking pulses.