There are automatic shotguns in limited use in the military, including the USAS-12 and the Franchi SPAS-15. These are rapid-fire, high-impact weapons, allowing the shooter to fire up to four shots per second with one pull of the trigger. The USAS-12 uses a drum magazine, and the SPAS-15 uses a box magazine.
Even more powerful is the Pancor Jackhammer, currently just a concept and prototype weapon. It's an automatic, drum-loaded shotgun made out of plastic. The Jackhammer is extremely light and has a remarkably small recoil. Most of the recoil energy is captured and used in loading and firing the next round. As an interesting additional feature, it is possible to take the drum magazine off the gun, attach a detonator and use it as an anti-personnel mine that fires all of the cartridges at once when tripped.