GODS OF THE UNDERWOLRD:
Hades 
(Lat. Pluto; Dis; Orcus)
= Aidoneus (Invisible); Pluto (Rich). Brother of Zeus (or Lower Zeus). 
Somber; rich; hospitable. Seldom appears above the ground and dislikes sunlight.  His epithets: Polyonymos  Polydegmon = of many names; host of many. Abducted Persephone for marriage.
Persephone 
(Lat. Proserpina)
Maiden/wife: goddess of spring; queen  of the dead. Fair and horrible at the same time.
Thanatos (= Death)  Winged daemon, snatches the life away. 
Hypnos (= Sleep)
also, Morpheus
and the numerous progeny of dreams.
Brother of Thanatos. His symbol: poppey, an opiate. 
Hermes Psychopompus 
(the Guide of Souls) 
Leads the souls to (and from) the Underworld. 
Charon Ferryman of death. After funerals, takes the souls across the river Styx.  Those who have had no proper rites are denied the passage and must endlessly roam between the worlds.
Erinyes (euphemistically -
Eumenides,Well-Disposed)
 Goddesses of curse and vengeance. Chase parricides and avenge atrocities by driving to insanity.
Roman: Manes 
guided by the goddess Mania
Ancestral spirits. If angry, drive the guilty descendant insane. 
Ker (Destruction); 
Lamia and Empousa
Vampires, snatch children.
Hecate (Lat. surname 
Trivia = of the Crossroads)
Three-headed goddess of witchcraft  in chariot of dragons. 
Appears at night on the crossroads with the torch and dogs.
Moirae (Lat. Fates) 3 sisters: Clotho pulls the life-thread; Lachesis weaves the fortune; Atropos cuts the thread.
Even gods fear them. They are really in control of everything. 
Judges of the dead: 
Minos, Radamanthus, Aeacus
Decide the post-mortem punishments or  rewards. 
Cerberus 3-headed dog. Guards the gates of Hades.

WATER OF DEATH:

Styx (Abhorrent)  River and goddess: oath of gods; renders invulnerability. 
Acheron (Contrition); Cocytus (laments); 
Phlegeton (or Pyriphlegeton: pyre) 
Rivers in the Underworld.
Lethe (Oblivion) Lake of amnesia. Drinking from it, souls forget their past life.

THE REALM OF HADES:

Hades Both god and the place of the dead. - Gloomy realm of incorporeal existence. 
Elysium
(Elysian fields)
Pleasant abode of the blessed souls.
Tartarus Fiery bottom of the universe, place of eternal punishments of great transgressors
and confinement of rebellious Titans. 
What do the dead people do? * Eat! Hades - hospitable god; food of the dead/ pomegranate; 
   Theseus & Perithous are seated at his eternal banquet. 
   (cf. traditional stelae - gravestones, representing the deceased at a banquet.)
* Fly around, endlessly blown in a listless whirlwind.
   Achilles in the Odyssey would exchange his prominent status among the 
   dead for the life of a poor farmer's hireling above.
* The dead know the past and future, but are unaware of the present.
* Some lucky have a good time in Elysium. 
* Some few serve the eternal time of punishment 
   (Tityus; Ixion; Tantalus;  Sisyphus; the 49 daughters of Danaus.)

CATABASIS - DESCENSUS AVERNO - DESCENT TO THE UNDERWORLD:
The theme of visiting the realm of death is central to epic tradition.

Why do alive people go to the realm of the death? * For advise/ prophecy - Odysseus (Nekyia); Aeneas. 
* To bring somebody or something from the realm of the 
   dead. (Gilgamesh; Orpheus; Heracles, et al.)
Gilgamesh Mesopotamia, 2500 B.C. - In search for immortality. He only manages to bring a good advise from his divine ancestor Utnapishtim, survivor of the flood. 
Odysseus Homeric epic, 900 B.C.: sent by Kirke (the goddess of metempsychosis?) to inquire of the way home.
Heracles Brings Cerberus; returns Theseus; fights Thanatos and returns Alcestis.
Theseus & Peritous In hope to re-abduct Persephone. Invited to join for dinner, got stuck to the rock seat. 
Orpheus For Eurydice: fails by looking back. 
Psyche On the demand of the mother-in-law, Aphrodite, to fetch a bit of Persephone's beauty.
Aeneas In Vergil (Rome ~10 B.C.) - to visit his father, who shows him the future of Rome. Vergil had influenced the vision of Dante, Inferno, A.D. 1265 - the central piece of European literature.
Odin Old Norse, ~A.D. 500. He awakes the dead Volva = Nordic Sibyl; she prophecises the doom of gods and the Ragnarok.
Mesopotamian goddess Istar; Persephone; Orphic Demeter Temporary confinement in the realm of death with subsequent return.
Dionysus Awakes mother Semele from her grave; saves and marries Ariadne (god-savior).
cf. Jesus Descending to the Underworld to release the souls of the patriarchs ("the harrowing of hell").
The purpose of Orpheus; Heracles; Demeter and Dionysus' catabasis is to release the souls from death: the prime focus of a mystery-religion.
 

PHILOSOPHY OF DEATH:

Pythagorean doctrine (~VI c. B.C.): metempsychosis - transmigration of souls; reincarnations. >Platonic views (IV c. B.C. + Orphism > Neo-Platonism > Gnosticism ~A.D.) Platonic goal: escape the wheel of reincarnations and re-join the proper and original deity of one's soul, from which she had split off due to some fault that caused her fall into matter.