Generally, immortality is the concept of living for an indefinite length of time, presumably in perpetual health, unless suffering a violent death.
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In the works of J.R.R. Tolkien, only beings of Arda such as the Ainur and Elves are immortal. The state of Orcs' lives is unknown. Men, Dwarves, Hobbits, and most other races cannot live forever and thus are subject to aging and natural death; however various members of those races have endured longer extended lifespans beyond usual constraints through multiple means.
In Arda, despite many having a desire for it, immortality is not desirable for those who are of mortal kind. Each race has a set span; to exceed this span brings agony for mortals. There are many examples of this: the kings of Númenor who became witless and agonized by attempting to cling to life as long as possible (as opposed to earlier kings who gave up their lives willingly); Gollum's transformation into a twisted and starved being after his life was stretched five times longer than natural, by the One Ring; Bilbo Baggins' comment on his 111th birthday about feeling "thin and stretched," like "butter being spread on too much bread," this also is an effect of the One Ring. The only time mortals have ever been accorded peace in immortality is when they are accounted of the Elves: this choice has been mostly restricted to the Peredhil, the Half-elves (the one known exception to this being Tuor, who was born a full-blooded Man but was given a choice to be accounted among the Eldar).
The Army of the Dead was cursed by Isildur with immortality in the form of undead spectres after they abandoned their oath in the War of the Last Alliance. They were doomed to never die until properly fulfilling their oath.
In the latter half of the Second Age, Sauron's corruption of the Númenórean King Ar-Pharazôn made Ar-Pharazôn believe that if he could invade and conquer the Blessed Land, he could become immortal. The vain attempt at this, in SA 3319, resulted in the destruction of Númenor by the Valar, and Ar-Pharazôn's imprisonment in the Caves of the Forgotten.