The heroic journey begins with the “call to adventure” which is made to a reluctant hero. Odysseus undergoes his a 10-year journey against his will. (kind of. See below.) If he had his druthers, he’d have happily gone straight back to Ithaca after the Trojan War. (Which is not to say he didn’t take some long vacations along the way — e.g., spending one year with Circe and five with Calypso.)
Once the hero embarks on his journey, he enters the “realm of adventure.” In this realm, the hero encounters things he ordinarily wouldn’t: Circe’s magic, the island of Calypso, the Cyclopes, the Phaeacians, etc. In the realm of adventure, the hero will have helpers such as Athena and Nausicaa and face hindrances such as Polyphemus, the Sirens and Poseidon.
One requisite part of the heroic journey is a trip to the Underworld. In Greek myth the hero literally visits the underworld. Odysseus did it; Heracles did it; Theseus did it. In modern iterations of the heroic journey such as “City Slickers” and “Joe versus the Volcano” the hero more realistically faces the possibility of his own death.
Eventually the hero returns from the Realm of Adventure to the “real world.” Upon arriving in the real world, the hero will find trouble there, as well. The final task of the hero on his journey is to overcome this trouble. His adventures up to this point will have been such that they help the hero set things right back home. In Odysseus’ case, he learns from Agamemnon’s negative example to return home in disguise before announcing his presence. He has 108 suitors to deal with — he can’t just barge in, guns blazing, as it were.
While in disguise, Odysseus recruits the help of his son, Telemachus. Together, with the help of Athena, they are able to defeat the suitors who have (more of less) taken over his household. It is important to note that Telemachus undertakes his own heroic journey in Books 1–4. It is only after his journey that he becomes capable of helping Odysseus fight the suitors and regain control of Ithaca.
