What are the challenges that Odysseus had to face on his journey home?
This is a good question. However, there are too many problems to mention in any detail. I will give you a brief summary of the difficulties that Odysseus had to face to make it back home to Ithaca.
First, Odysseus is a captive of Calypso. He is stranded on her island and cannot go home. However, Zeus intervenes and forces her hand to allow him to leave. As he leaves, Poseidon does not like him very much and almost kills him. He barely makes it to the Phaeacians.
While he is with the Phaeacians, he hears the story of the Trojan War and he begins to weep. After he calms down, he recounts his hardships. This is where we hear of his travels. First, he is on the land of the lotus-eaters, where some of his men eat the fruit of the lotus, which makes them forget. He barely makes them go back to their ships. Then his crew is almost eaten on the island of Cyclopes. Only through his cleverness do they escape. Also on his journey he also had to face Circe, the witch, and other hardship on the sea, such as Scylla and Charybdis. He loses men in the process. Finally, the men gain the ire of Zeus and they all die, as Zeus strikes their ship. Odysseus is the only survivor.
When the Phaeacians hear of these tales, they decided to help him. He finally arrives at home, and he realizes that it is overrun with suitors. He has to face them as well. He eventually kills them all. In short, Odysseus faces many hardships. He is a veritable man of sorrows.