One of the important themes in the Odyssey is that of xenia, a mixture of guest friendship and hospitality. Thinking about it properly, it's clear that this theme also has parallels in the Hobbit. In the Odyssey, the encounter with the Cyclops is seen as the worst example of hospitality, and there are links between this scene and those in the Hobbit. For example, the trolls, like the Cyclops, threaten to eat the adventurers, while the elvenking Thranduil imprisons them in his underground caves. We can also perhaps see a similarity between Odysseus' 'nobody' trick which he uses to escape, and Bilbo's invisibility granted by the ring. While they are not collected together as they are in the Odyssey, several elements of the Cyclops scene are present in the Hobbit as well.
Another major theme of the Odyssey is that of nostos ('homecoming'). As well as the homecoming of Odysseus, the nostoi of Agamemnon and Menelaus are also looked at in detail, and homecoming is also a theme represented in the Hobbit. The entire storyline of the book revolves around the dwarves attempt to return home, and this journey home is comparable to the wanderings of Odysseus (although much shorter - pun possibly intended). There is also a second homecoming in the Hobbit, when Bilbo returns to the Shire. In both of these nostoi, another element of this theme in the Odyssey is reflected, that of troubles at home. When Odysseus returns home there are suitors in his house, consuming his estate and not expecting him to return. Likewise, when Agamemnon returns he is killed by his wife Klytaimnestra, and Smaug makes attempts to kill the dwarves when they return.
Finally, kleos (a difficult word to translate, perhaps best rendered as 'reputation among others') is another important theme of the Odyssey which can also be found in the Hobbit. Like Odysseus, the dwarves gain kleos through reclaiming their home, and this is particularly well shown in the Hobbit films - Thorin is obsessed with reclaiming the Arkenstone because it will effectively allow him to regain kleos among the other dwarf lords. Another instance of kleos in the Odyssey is that acquired by Telemachos on his travels. By travelling and meeting others, he increases his reputation among others. We can see that the opposite is true to some extent in Bilbo's case. While he does earn respect among the dwarves, his reputation at home is weakened because of his adventures.
There you have it! It's been interesting to compare the Odyssey and the Hobbit, and I've come to the conclusion that there are a good number of thematic similarities between them. However there are also a good number of differences, such as the details of hospitality and kleos (or lack thereof) of Bilbo on his return from his adventure. I've found this to be a very interesting way to look at the Hobbit, and it's good to see it from a thematic perspective - something I've never really considered before.
Thank you for reading, and feel free to discuss below (or to mention something I've missed). This might be my last post before Christmas (though I hold out hopes for another, since I'm now home from uni for Christmas). If so, then I wish you all a very happy Saturnalia!