Odysseus shows bravery many times throughout the course of the Odyssey. Here are a few of those examples.

During his encounter with the Cyclops (Polyphemus), Odysseus and his men get stuck in a precarious situation. Not only does Odysseus get them stuck in Polyphemus's cave, many of his men die a violent death at the hands of the creature due to Odysseus's hubris. But Odysseus keeps his nerve and devises a plan to escape. He offers the creature wine, stabs it in the eye, and risks death riding under the bellies of the ram to escape from the cave.

Odysseus is brave when he travels to the Underworld to hear his prophecy told by Tiresias, a famous prophet. Not only must Odysseus fend off the dead from the sacrificial pool, he learns that his mother has passed away while he's been at war. He doesn't let either of these scenes shake him, solely focusing on the prophecy that will save his life and get him home.

As Odysseus approaches Scylla and Charybdis, fear begins to permeate his courageous mind. He knows the fate of his men if he chooses this path, but he also knows it's the only way to get home to his family. Odysseus watches as Scylla swoops down and snatches up six of his men and drags them back to her cave as they scream for help. As he moves away from her and toward Charybdis, he recalls this being one of his darkest moments, for he can do nothing to help.

When Odysseus makes it back to Ithaca, he finds over a hundred suitors living in his home. With the help of Athena, he devises a plan to win a contest that distracts the men. He then works with his son, Telemachus, and a few servants to overthrow the suitors and return his home to peace. Fear is never on his mind. His sole focus is to protect his family and win back his wife, Penelope.

In all of these situations, Odysseus may feel some fear, but he never lets it overtake him. His bravery shines during difficult moments, illuminating the traits of a true epic hero.

There are numerous moments throughout Odysseus's treacherous journey home when he displays a tremendous amount of bravery and courage. Odysseus demonstrates bravery by formulating a plan to blind Poseidon's son, Polyphemus, and cleverly escaping his cave. Odysseus risks his life to get Polyphemus drunk and stabs him in the eye with a massive spike before leaving the cave on the underbellies of Polyphemus's flock of sheep. Odysseus also demonstrates bravery on Circe's island by volunteering to save his men who have been turned into pigs by the powerful goddess. Fortunately, Odysseus is able to avoid Circe's charm with a magic antidote from Hermes and saves his men.

Odysseus also displays bravery by sailing to Oceanus and speaking to the terrifying shades of death in the underworld, where he receives instructions from the blind seer Teiresias. Odysseus once again demonstrates bravery by maintaining his composure when he sails past the monster Scylla and the menacing Charybdis. One could also recognize Odysseus's bravery during his final battle with the suitors. Despite being outnumbered, Odysseus and his son defeat the unscrupulous suitors and restore peace to his home.

One of the many examples of Odysseus's bravery comes in the episode with the Sirens. The Sirens are dangerous creatures who lure sailors to their deaths with their beautiful, golden voices. Circe has already warned Odysseus of the dangers of listening to the Sirens' song. Odysseus understands this, but at the same time, he is curious to hear their sweet, seductive melody for himself.

So he devises a plan. He orders his men to tie him to the ship's mast and to not release him under any circumstances, no matter how much he begs them. On Circe's suggestion, Odysseus's crew will have their ears plugged with beeswax, so they won't be able to hear the Sirens' song. Yet, tied firmly to the mast, Odysseus will hear every note, suffering the agony of desperately wanting to escape while being unable to do so.

Odysseus displays his bravery in The Odyssey when he travels to the Underworld in order to seek out Tiresias, the blind prophet who can tell him how to get home.  As he begins to see the spirits approaching the blood of the sheep they'd sacrificed, he says that "Pale terror seized [him]."  However, he knew that his men would need to see him be strong, and he felt the responsibility to "inspir[e]" them, and so he ordered them to deal with the sacrificed sheep's bodies while he took up his sword and held off the shades until he could speak with Tiresias.  The thought of facing the spirits of so many dead people is absolutely frightening, and Odysseus -- brave as he is -- feels the terror of being in this situation.  However, he bravely faces his fears and attempts to inspire his crew because he knows that they need information that only Teiresias can give.  Despite his terror, he stays and accomplishes what he set out to do.

As the hero of Homer's famous poem, Odysseus is more or less constantly performing acts of heroism and bravery. However, there are a few exceptional examples of Odysseus' bravery that are worth dwelling on. The Cyclops episode, for instance, is perhaps one of the best examples of Odysseus' legendary courage. 

Odysseus meets Polyphemus the Cyclops when the monstrous beast traps him and his men in a cave and threatens to eat them alive (only after, of course, he actually makes good on this promise and stuffs several of them into his mouth). However, while the Cyclops is sleeping, Odysseus summons the courage to sharpen a wooden stake and use it to gouge out the monster's eye. Odysseus' bravery rescues his men from certain death (although they ultimately meet their end in other equally grisly ways). 

While Odysseus shows his bravery in facing the Cyclops, however, he also displays his supremely troublesome arrogance. While he is rowing away from the island, Odysseus taunts Polyphemus, prompting the monster pray to his father, Poseidon, and doom Odysseus to many years of toiling and wandering. As such, even if Odysseus is undeniably brave, his is also undeniably foolish and arrogant.