Odysseus faces a number of challenges on his way home, including:

• sailors captivated by the lotus-eaters
• a battle with Polyphemus, a cyclops
• a storm sent by Poseidon
• an encounter with a witch
• being captivated by the Sirens
• encounters with the monsters Scylla and Charybdis
• punishment by Zeus 

On the way home from war, Odysseus's soldiers are captivated by the lotus-eaters, who try to feed the men fruit that will cause them to forget their desire to go home.

Next, Odysseus and his men get trapped in the lair of Polyphemus, who eats some of the men. Odysseus blinds Polyphemus, and the cyclops calls out to his father Poseidon, who sends a storm to punish Odysseus. Later, Odysseus refers to the encounter with Polyphemus when he tries to encourage and bolster his men, saying, 

Dear friends, surely we are not unlearned in evils. This is no greater evil now than it was when the Cyclops had us cooped in his hollow cave by force and violence, but even there, by my courage and counsel and my intelligence, we escaped away. I think that all this will be remembered some day too. Then do as I say, let us all be won over.

Odysseus's ship travels to the island of a witch named Circe, who turns many of his sailors into swine. He goes out to find an herb that counteracts her magic and convinces her to turn his men back into people. After that, Odysseus and his men choose to stay on the island for a long time.

Once it is ready to continue the journey to Ithaca, the crew is forced to sail by the land of the Sirens, beautiful monsters who sing to attract ships toward rocks where they will be smashed. The men plug their ears and tie Odysseus to the mast to help avoid the danger and temptation. They also avoid the whirlpool of Charybdis, but this forces them to sail near Scylla, who lives across from Charybdis. Consequently, six men are killed.

On the island of Thrinacia, Odysseus's men hunt sacred cattle that belong to Helios, the sun god. Helios demands his father Zeus to punish Odysseus and his men for the action.  Zeus obliges, sending Odysseus's ship into Charybdis. Odysseus is the only survivor. He says, 

At this time Charybdis sucked down the sea's salt water, but I reached high in the air above me, to where the tall fig tree grew, and caught hold of it and clung like a bat; there was no place where I could firmly brace my feet, or climb up it, for the roots of it were far from me, and the branches hung out far, big and long branches that overshadowered Charybdis. Inexorably I hung on, waiting for her to vomit the keel and mast back up again. I longed for them, and they came late; at the time when a man leaves the law court, for dinner, after judging the many disputes brought him by litigious young men; that was the time it took the timbers to appear from Charybdis.

He spends the next seven years as the prisoner and lover of Calypso. He is only able to continue his journey home when Athena intervenes with Zeus and convinces him to free Odysseus.

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.