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.