Shape Shifters

Many creatures exist within Middle Earth that seem able to alter their form to that of an animal, and there are hints throughout the Lord of the Rings and the Silmarillion that this is possible. This ability is never fully explained, and as such, is often overlooked or ignored.

The Valar
The Valar are not constricted to any one form, but often choose to take forms similar to the children of Illuvatar. It says of this in the Ainulindale: "Their shape comes of the knowledge of the visible world, rather than of the world itself; and they need it not, save only as we use raiment...Therefore the Valar may walk, if they will, unclad, and then even the Eldar cannot clearly perceive them, though they be present. But when they desire to clothe themselves the Valar take upon them forms some as of male and some as of female."

Beorn
One shape shifter that we do meet in the Hobbit is Beorn. Beorn is a very mysterious character, who lives alone in a house near the Carrock (as he calls it). He is introduced by Gandalf as "A very great person", and a Skin Changer, which seems to have similar meaning as Shape Shifter, as he is able to change between a man and a bear. Little is known about his origins, but Gandalf says of him: "Some say that he is a bear descended from the great and ancient bears of the mountains that lived there before the giants came. Others say that he is a man descended from the first men who lived before Smaug and the other dragons came into this part of the world, and before the goblins came into the hills out of the North. I cannot say, though I fancy the last is the true tale. He is not the sort of person to ask questions of."

Beorn lived with many animals such as horses and dogs, as well as keeping bees bigger than hornets. These animals were seen to be very intelligent, able to understand Beorn's words, and the dogs were able to walk on their hind legs and carry things in their front paws. It is unknown whether these were skin changers as well ( they certainly don't seem to be normal animals.)

Sauron
Although nothing is mentioned of this in the Lord of the Rings, in the Silmarillion Sauron is known as Lord of Werewolves, and is seen to be able to change into the form of a werewolf to battle with Huan: "Therefore he took upon himself the form of a werewolf, and made himself the mightiest that had yet walked the world." During the fight he is also seen to change into other forms: "Then Sauron shifted shape from wolf to serpent, and from monster to his own accustomed form, but he could not elude the grip of Huan without forsaking his body entirely."

Speculations
It may be that Maiar, being lesser versions of the Valar, also share this ability to change their appearance at will. It is not stated that the Maiar were different beings to the Valar entirely, in fact it states in the Silmarillion regarding the Maiar: "With the Valar came other spirits whose being also began before the World, of the same order of the Valar but of less degree." This would certainly explain Sauron's ability to change at will. There is also evidence to suggest that other Maiar possess this power:

Radagast
During the Council of Elrond Gandalf refers to Radagast as "A master of shapes and changes of hue." This would seem to imply that Radagast is also able to change his form.

Gandalf
The Silmarillion does not speak much of Olorin (Gandalf) but this it does say: "For though he loved the Elves, he walked among them unseen, or in form as one of them." This would seem to suggest that Gandalf was able to change his form at least to that of an Elf. However, this ability may seem inconsistent to the Hobbit and the Lord of the Rings, and it may be asked why, if he had this power, he did not use it more often. Nevertheless, as this ability is never explained, only hinted at, there may be unknown limitations to this that would explain why it is not more commonly used.

Common Misconceptions
It should be noted that Werewolves are not shapeshifters; they are limited to only their wolf form, unlike some non LOTR legends where they become men.