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Many stories about the Greco-Roman Underworld mention the rivers flowing through it: the Styx, the Acheron, the Lethe, the Pyriphlegethon, the Cocytus, the Eridanus, and sometimes others.
Sources differ about which ones are actually rivers, though. For example, book six of the Aeneid describes Acheron as being a whirlpool, and Styx the marsh around it.
Which other sources specifically discuss or describe the rivers of the Underworld?
The description of Acheron and Styx in Aeneid 6 appears to be fairly clearly based on the somewhat ambiguous Underworld structure supplied by Circe in Homer's Odyssey 10. Other mythography subsequent to Homer likewise appears to take Circe at her word as far as the placement of these chthonic features is concerned.
E.g. Plato's interpretation, given in Phaedo 112e-133c, is that "Tartarus" (here apparently a generic term for the entire Underworld, as is the case in other writers, such as Hyginus, for instance) contains four main streams (rheûmata), namely Oceanus (which encircles the earth and flows in the upper world as well), Acheron, Pyriphlegethon and Cocytus.
Phaedo explicitly differentiates between a rheûma (stream) named Acheron and a límnē (lake) called Acherusia. Like Oceanus, Acheron flows also in the upper world, itself going "through various desert places and, passing under the earth, comes to the Acherusian lake." Virgil seems to retain a similar distinction, and perhaps paralleling with his Aeneid Underworld construction, in Phaedo, Plato says that Styx is a lake which the Cocytus drains into. Somewhat puzzlingly, according to Phaedo's report of the Cocytus, "it is that which is called by the poets the Stygian river" (because it forms into Lake Styx?).
Circe seems to describe an opposite flow in the Odyssey, saying that the Cocytus gets its water from the Styx, of which the Cocytus is thus a tributary. It doesn't seem expressly clear in the Odyssey what exactly Styx is, and so presumably whence the license for Plato to describe it as a lake, and for Virgil as yet some other sort of water-body. Meanwhile in Seneca's Hercules Furens, the Acheron and the Styx are two branches flowing from a single source. Here Styx is silent and placid while Acheron "with mighty roar rushes fiercely on, rolling down rocks in its flood... that cannot be recrossed."
Pausanias describes his visit to the northeastern region of Greece called Thesprotia in his Description of Greece 1, wherein he supports Plato's distinction between the River Acheron and Lake Acherusia:
Near Cichyrus is a lake called Acherusia, and a river called Acheron. There is also Cocytus, a most unlovely stream. I believe it was because Homer had seen these places that he made bold to describe in his poems the regions of Hades, and gave to the rivers there the names of those in Thesprotia.
Confirming both Plato and Pausanias, the 10th-century AD encyclopaedia called the Suda contains separate article entries for Acheron and Acherusia. It defines Acheron as a "river in Hades {potamós en ā́dou} mentioned in myth", while "Acherusia is a lake in Hades {límnē en ā́dou}, which the dying cross over". Styx meanwhile is described in this work as a spring (krḗnē) or fountain (pēgḗ) in the Underworld.
In Phaedo, at some point the fiery stream of Pyriphlegethon "pours into a vast region of fire, and forms a lake larger than the Mediterranean Sea, boiling with water and mud," and it too, after proceeding "muddy and turbid," winds around the earth, coming eventually to drain into Lake Acherusia.
In Theogony 775-806, Hesiod describes the structure of the earth and the netherworld in relation to Oceanus and Styx, telling us that one tenth of all the water of the cosmic stream of the Titan Oceanus is allotted to his daughter Styx, though he does not explicitly say that Styx herself is a river. Perhaps, though, he doesn't need to, since, at least in the world of the living, Styx was well-known as a river that flowed down a mountain in Arcadia, with a waterfall mentioned by Pausanias in The Description of Greece 8, where we are reminded that Homer has the same stream run down into the land of the dead.
Aeneid 6 also tells of the River Lethe, the stream of "Oblivion" from whose waters the dead who are destined for reincarnation drink in order to empty themselves of their previous lives' memories. Orphic Hymn 84 to Hypnos alludes to this river as well. In Plato's Republic 10 this river is called Ameles, "Unmindfulness."
One source mentioning the rivers is Book X of the Odyssey:
So I spoke, and the beautiful goddess straightway made answer: ‘Son of Laertes, sprung from Zeus, Odysseus of many devices, let there be in thy mind no concern for a pilot to guide thy ship, but set up thy mast, and spread the white sail, and sit thee down; and the breath of the North Wind will bear her onward. But when in thy ship thou hast now crossed the stream of Oceanus, where is a level shore and the groves of Persephone—tall poplars, and willows that shed their fruit—there do thou beach thy ship by the deep eddying Oceanus, but go thyself to the dank house of Hades. There into Acheron flow Periphlegethon and Cocytus, which is a branch of the water of the Styx; and there is a rock, and the meeting place of the two roaring rivers. Thither, prince, do thou draw nigh, as I bid thee, and dig a pit of a cubit's length this way and that, and around it pour a libation to all the dead, first with milk and honey, thereafter with sweet wine, and in the third place with water, and sprinkle thereon white barley meal.
