In Greek mythology, Charon (or Kharon) was the ferryman who carried the souls of the dead into the underworld, across the Acheron river. In Roman mythology, he carried them across the river Styx. He was the son of goddess Nyx and of the primordial deity Erebus.
He was represented as an old and ugly man, with a dirty beard and a crooked nose. In Roman poetry, he looked even worse - Virgil portrays him as a filthy man, with an unkempt beard and eyes burning like fire, dressed in a dirty tunic and having a greasy girdle. Also, in these more recent representations, he is ill-tempered and scolds the shadows of the dead all the time.
Charon the ferryman was usually represented in art in his boat, with one hand on the steering pole or on an oar and the other hand resting on the boat or inviting the shadows to come. The souls of the deceased were guided by Hermes, who was also a psychopomp ("guide of the souls").
For his services, he was paid the so-called "Charon's obolus" - that's why dead people were buried with a coin in their mouth or on it (an obolus or a danake). Several ancient towns were exempt from this "tax", either because they had been kind to a god, or because they were considered "shortcuts" to the underworld, so there was no need to pay for such a short trip (ancient sarcasm, if I may say so!).
The people who didn't receive a proper burial and thus had no obolus were left to wander on the shores of the river Acheron for one hundred years. That's why for the ancient Greeks it was so important to give a respectable burial to the deceased. For instance, Antigone risked her life by burying her brother, Polynices, because he was considered a traitor and the king of Thebes forbid everyone to bury him.
Plato tells us that the souls of the deceased were judged and then, according to their sins, they were taken by Charon to different areas where they were purified from their sins or where they received punishment.
Charon also had to ferry some living people, even if he didn't really want to. Among them were the hero Heracles, who easily convinced Charon by using his club, and Orpheus, who probably convinced him with his song, when he got into the underworld. After returning to the world of the living and looking back, he lost Eurydice for the second time. Orpheus could see her taken by Charon to the other shore; as much as he wanted to return to Hades, Charon refused to ferry him again.
Psyche went to the underworld sent on an errand by Venus; she took with her two coins, so as to pay Charon her way back, too.
In Roman literature, in Virgil's Aeneid, the hero Aeneas descends into the afterworld, accompanied by the Sybil, who shows Charon a golden bough, in order to convince him to let the hero across the river. (I liked it how, when a human boards the boat, as he is very heavy, the water gets in between the boards).
Last, but not least, Charon also appears in Dante's Inferno (from the Divine Comedy), as a demon with fire eyes, who beats with his oar the souls who don't get on his boat fast enough.
Here are some pictures of Charon:
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Many of the paintings and engravings are inspired from Dante's verses and show the angry Charon hurrying the souls.