The regenerative circuit (or self-regenerative circuit) allows a signal to be amplified many times by the same tube, and improved selectivity as well. This is also referred to as the "heterodyne principle".
A Tesla coil is a high-voltage, air-core, self-regenerative resonant transformer that generates very high voltages at high frequency, named after its inventor Nikola Tesla. This coil is part of Tesla's wireless transmission of electric power distribution system (US1119732 - Apparatus for Transmitting Electrical Energy - 1902 January 18). In modified form, this type of transformer, called a flyback transformer, provides the voltage needed to power the cathode ray tube that makes (non-LCD) televisions and monitors possible.
Regenerative circuits were also a great milestone in radio history. The "inventor" of FM radio, Edwin Armstrong, patents a method of regenerative circuits (invented while he was a junior in college, and patented 1914), the Super-regenerative circuit (patented 1922), and the Super Heterodyne receiver (patented 1918). Lee De Forest filed a patent in 1916 that became the cause of a contentious lawsuit with the prolific inventor Armstrong, whose patent for the regenerative circuit had been issued in 1914. The lawsuit lasted twelve years, winding its way through the appeals process and ending up at the Supreme Court. The Court ruled in favor of De Forest, although the experts still disagree about whether the correct judgement had been issued.
The secret to the regenerative radio was carefully controlled positive feedback. This also allowed the receiver to oscillate, allowing CW (morse code) to be heard as beeps. The regenerative receiver is theoretically as sensitive as any radio can be, however, the adjustments are critical, and must be continuously monitored during listening. If misadjusted the receiver can transmit interference.
The superregenerative receiver uses a quenching oscillator to permit high positive regeneration of the radio amplifying stage, while quenching or keying the built up regenerative oscillation at an untrasonic rate. This further improved the gain of the receiver while simplifying adjustment.
The regenerative radio made the most out of very few parts. When the parts became easier to obtain, the superheterodyne receiver replaced it for all serious work. The superheterodyne receiver is the most common receiver in use today.