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The Eagles were immense flying birds that were sentient, and could speak.
The eagles are said in The Silmarillion to have been "devised" by Manwë Súlimo, leader of the Valar, and were often called Eagles of Manwë. They were sent from Valinor to Middle-earth to keep an eye on the exiled Ñoldor, and on their foe the evil Vala Morgoth.
For a time the Lord of the Eagles, Thorondor, kept his eyries at the top of Thangorodrim, the volcano above Angband itself. While they lived there, Thorondor helped Fingon rescue Maedhros. Thorondor's folk later removed their eyries to the Crissaegrim, part of the Echoriath or Encircling Mountains about Gondolin. There they were friends of Turgon, and kept spies off the mountains.
Thorondor wounded Morgoth in the face after Morgoth's battle with Fingolfin, and he carried Fingolfin's corpse to the Echoriath, where he was buried by Fingon.
In the Second Age, a pair of Eagles had an eyrie in the King's House in Armenelos, the capital of Númenor until the Kings became hostile to the Valar. The Eagles also watched the peak of Mount Meneltarma, and three Eagles would always appear when someone climbed to the summit.
In The Hobbit, no eagles are identified by name. Only the title Lord of the Eagles distinguishes the eagle leader from other eagles in this story. The text adds that he was given the title King of All Birds at a later date. Many readers assume, without hard evidence, that it was Gwaihir and Landroval who rescued Thorin Oakenshield and company from a band of Wargs and Goblins, flying them to the river Anduin, and later assisted in the Battle of Five Armies fought near Erebor, the Lonely Mountain.
Before and during the War of the Ring, one of Gwaihir's people rescued Gandalf the Grey from the top of Isengard, and others of his people rescued Frodo Baggins and Samwise Gamgee from Mount Doom in Mordor after the One Ring had been destroyed.
Tolkien's painting of an eagle on a crag appears in some editions of The Hobbit. According to Christopher Tolkien, the author based this picture on a painting by Archibald Thorburn of an immature Golden Eagle, which Christopher found for him in The Birds of the British Isles by T.A. Coward. However, Tolkien's use of this model does not necessarily mean that his birds were ordinary Golden Eagles. In his last writings Tolkien speculated that these great Eagles were actually Maiar in bird-shape, as he felt it unlikely Ilúvatar would grant feär to animals. If this is true, then Roäc the Raven and the Thrush, who appear in The Hobbit, must also be Maiar or other spirits in animal form (and possibly even Beorn, who sometimes takes the form of a bear).
Flying the Ring to Mount Doom
Many skeptics have asked why the Eagles simply didn't carry Frodo and the One Ring into Mordor and drop the Ring in Mount Doom. At first glance this seems incredibly easy compared to the alternative but there are a few reasons which prevent this idea.
Firstly, the Eagles would most likely have become corrupted by the power of the Ring and would have most likely attempted to prevent the destruction of the Ring. But this can be contended with the argument that Gandalf knew that anyone might and would refuse to throw in the Ring.
Another objection is that the Eagles coming from the air would have been fairly obvious to Sauron and his spies, and the Fell beasts would most likely have stopped the attempt. Also, it is probable that Mordor's numerous archers would create great trouble. The distances, greatly decreased by the movie, are in the book quite large. Mordor being full of orcs, they would be under constant fire, unless they could maintain a great height for a long period of time. The Eagles expressed fear in The Hobbit about going into the Lands of Men because of their bows. This, combined with the Fell Beasts, would probably not only hinder but completely wipe them out.
Another proposition is that the Eagles would have refused to aid the Fellowship in destroying the Ring because they, being emissaries of the Valar like Gandalf, were not allowed to go on the offensive against evil. Flying the Ring to Mordor could have been the Fellowship's first priority but maybe they were unable to contact the Eagles in time.
Lastly, and perhaps most obviously, many state the reason for the Eagles not flying it to Mordor is simply because it would have made a boring book. Few would disagree with this.
The party of Tolkienists that accepts this as a plot hole usually responds that in any book there are usually plot holes. In a larger, far more detailed and realistic book we expect fewer (if any) plot holes, when in reality there is a far greater chance.
It is interesting to note that for Peter Jackson's adaptation of The Lord of the Rings, many cast and crewmembers have casually joked about how "the Eagles don't take the Ring to Mordor because that would have ended the story quickly!". Particularly, on the writer-director DVD commentary track, Peter Jackson and Fran Walsh repeat this statement and begin joking around about it. However, writing partner Philipa Boyens then bursts out and angrily declares "Why does everyone always say that?! The flying Nazgul on their Fell Beasts would have stopped them! How more obvious does that need to be?! Mordor has flying creatures too!" As a consequence, Jackson and Walsh fall silent, then quietly admit that her explanation entirely makes sense.