The Cosmos & The Gods
One idea that makes Hellenic Paganism stand out from many religions is that, while Hellenes believe in many gods, we do not believe that the gods created the Universe, and many of us (myself included) do not believe the gods are omnipotent or entirely benevolent (although they are far more highly evolved than humans, and vastly more powerful and more benevolent). Most Hellenes also do not believe that the gods require or demand worship. They are worthy of our worship, and through a worshipful relationship with them we may experience many wonderful things (friendship, guidance and blessings), but they do not seek to punish those who do not worship them. The gods concern themselves more with our behavior than our beliefs. The gods will help a just man, regardless of his beliefs, but will punish an unjust man just as readily. Of course the fates and the laws of the Cosmos also have their say occassionally, and actions that please one god may displease another, so no amount of purity or goodness can ensure a good life--on the other hand, those qualities are good in and of themselves and certainly can't hurt. Thus, the relationship between men and gods is one of honest friendship but unequal power; the gods can be considered wise teachers, guardians and friends in [very] high places.
Most Hellenes honor all of the gods, and make a point to pay homage to each of the Olympian twelve at least once during each year (if not more often), but truly worship and are devoted to a small handful of patron and local gods. Thus, where the gods may differ on the virtue of an action (for example, Artemis is a chaste goddess, whereas Aphrodite is far from chaste) a human may relate more to one god, but for balance's sake will also honor the value of the other. In our hypothetical example here, a person may worship Aphrodite and enjoy their sexuality fully, but will also honor the beauty and place that abstinance has in the Universe and will ensure that they do not go to excess (of course, the reverse could also be used as an example). Many Hellenes also recognize and honor a variety of local or minor (demi) gods, as well as nature spirits (such as nymphs and dryads). Given the multiplicity we see in the divine, we recognize that occassionally a dilemma will arise where there is no right or best course of action, or where doing the right thing may cause us to suffer greatly. It is during these times especially when we honor our gods and do what we feel we must for ourselves and our family, friends and our community. Just like our gods, we Hellenes also tend to focus more on people's actions and less on their beliefs. Hellenism is a religion of orthopraxy (right-doing) rather than of orthodoxy (right-thinking). Beliefs on the Universe, the afterlife, morality and philosophy range wide and far now (just as they did in ancient times), but we are all one through our rituals and our mutual love for the gods. Consider this the standard Pagan warning of "ask five [Hellenic] Pagans a question and get six different answers." I will attempt to represent a variety of beliefs, but will focus largely on explaining my own--my ideas on this page (and the others on this site) are mine and do not necessarily represent those of all other Hellenes.
Who are our gods then? Our main gods are the Olympian Twelve, or Dodecatheon, who dwell on Mount Olympus. As with the rest of myth, Hellenes do not interpret this literally, but rather symbolically. Thus, we do not believe that there is a palace on top of Mount Olympus, but rather see Olympus as a spiritual realm--symbolized by the actual mountain. The Olympian gods are balanced, in that there are six gods and six goddesses. The most common version of the Dodecatheon includes Zeus, Hermes, Apollon, Ares, Poseidon, Hades, Hera, Aphrodite, Demeter, Athena, Artemis and Hestia. Each of these gods has many epithets, or sub-names, by which they are honored. Each epithet recalls a specific aspect of that god's existance, thus we may call upon the wisdom of Athena (Athena Glaukopis [owl/grey-eyed]) rather than simply calling upon Athena. Foreign (non-Greek) gods were often honored in ancient Greece, and the practice continues today. Except where such gods have specific Hellenic forms of worship, "foreign" gods are generally honored in the manner that their native culture honors them (or as close to that as possible). Minor, local and domestic gods, spirits and heroes are also commonly recognized, including the Agathos Daemon ("good spirit") or higher self, a concept similar to the Christian idea of a guardian angel.
If the gods are not all powerful and at the peak of the Universe, then, what kind of cosmology do we have? One popular cosmology is based on Neoplatonic and Orphean teachings. This Cosmology has, at its peak, The One (or The Cosmos) which is all knowing and all powerful. The Cosmos is the source of all things, all manifestation springs from it, yet The Cosmos itself is never made manifest. The Cosmos is beyond thought or action and is simply the whole of everything, the totality of universes. It is impersonal and its sole concern is with the balance of the Universe, not the petty lives and actions of gods and men. From The One springs The Divine Mind, the first thought and first true being. From this springs The Soul (aka. The World-Soul or All-Soul), each of our souls is a reflection of this One Soul and through it were are united. Matter itself springs from it as well. All things are destined to evolve back toward The One over time. Hellenes who take this view often call this goal Apotheosis, a word with meanining similar to the Buddhist term Enlightenment. Some Hellenes (myself included) believe in rebirth or reincarnation and feel that we evolve spiritually toward this goal through many lifetimes. There is an interesting treatment of this idea here.
Other than reincarnation or rebirth, then, what other concepts of the afterlife are found in Hellenism? Some Hellenes do not believe in an afterlife, or remain skeptical, feeling that our earthly existance is what is alloted to us and we must accept this. Others believe that we go on to another world after death, where we live eternally. Among those that share this belief, some believe that we go to the same afterworld, others believe that Tartaros (a place of punishment) and Elsyia (a place of reward) await those who are especially deserving, while a neutral realm (Hades) is the destiny of most. Many who believe in rebirth also believe that we may pass briefly through such realms in between lives, as we evolve toward Apotheosis.