Chopping Off Heads
Decapitation isn't nearly so rare or so "medieval" as many commentators apparently believe. My favorite Italian newspaper, il Foglio (Rome), recently published an excellent overview of the practice. If you read Italian, have at it. If you don't, here are the points that seem most useful to a contemporary person trying to understand what's happening, with some of my own thoughts on this ghoulish subject.
Decapitation was a well-established method of enforcing the death penalty throughout Europe until fairly recently. The French, to take the most recent example, only stopped beheading convicted killers in 1981 when Francois Mitterrand abolished the death penalty itself. The French method--the guillotine--was famous, having been established during the period of the Revolution, in 1792. It spread far and wide.
The guillotine--named after one of its designers, Joseph Guillotin--was a more humane method of execution than either hanging or beheading with a sword. Both of the latter methods were subject to technical failure (the gallows did not always break the neck of the victim, prolonging the agony for several minutes, and the "execution sword"--with a sharp tip in Asia and Africa, but a blunt end in Europe--didn't always do the job with a single stroke). The guillotine nearly always worked. It worked so well that it replaced previous methods, and was used to execute criminals from all social classes, kings to beggars.
Footnote: Guillotin wasn't the inventor; that honor goes to another Frenchman, Antoine Louis, who of course wanted the device to be called the "Louisette."
Another footnote: over its "career," the "widow" chopped off between fifteen and twenty-five thousand heads.
Yet another footnote: The most famous failure was the bungled execution of French King Louis XVI. It didn't kill him right away -- he suffered horribly, to the delight of the mob.
Back to the very beginning: The first systematic practice of decapitation was in Egypt at the time of the pharaohs, from the 4th millenium to the fourth century B.C. It was adopted by Imperial Rome for full-fledged citizens (slaves and thieves were crucified).
Today, the only country (not counting the Islamic State) that decapitates as an official method of execution is Saudi Arabia (however some NGOs claim that it is still used in parts of Africa and Asia). Terrorists do it regularly, as do Mexican drug cartels. Almost all the decapitating states used swords (the current use of knives, which takes longer, combines torture with execution).