"Believe me, captain, immortality consists largely of boredom."
Immortality was the avoidance or reversion of death or an entity that appeared to exist forever.
Immortality was one of the defining characteristics of the Q. The fact that the Q Continuum forced immortality onto other Q dismayed one of the Q, who attempted suicide. He claimed that immortality was one of the vulnerabilities of the Q. (VOY: "Death Wish")
The Greek gods were immortal. In order to end their existence, they spread themselves against the wind, becoming thinner until reaching a point where they could not reform into their bodies. (TOS: "Who Mourns for Adonais?")
The Douwd were immortal beings known for hiding their identity. (TNG: "The Survivors")
While living on Earth, Flint possessed immortality in the form of instant tissue regeneration. He lost his immortality when he left to live on Holberg 917G. (TOS: "Requiem for Methuselah")
Zefram Cochrane was rejuvenated and made immortal by the Companion in 2119. He gave up his immortality to grow old and die with the Companion. (TOS: "Metamorphosis")
This heavily implies that the Companion was immortal or near-immortal as well.
The Gideons, in practice, were immortal for awhile between the elimination and reintroduction of germs on their planet, as death came to them only when their body couldn't regenerate itself due to disease. (TOS: "The Mark of Gideon")
The Ba'ku planet released metaphasic radiation which caused cells of humanoids to be continuously regenerated, resulting in a form of immortality. (Star Trek: Insurrection)
Dr. Bathkin and his successor Elias Giger attempted to make themselves immortal by developing a cellular regeneration and entertainment chamber. Giger eventually succeeded in creating this machine and shared his interest with Weyoun. (DS9: "In the Cards")
Roger Korby believed that a person could have practical immortality by continuously transferring his or her consciousness into an android body in a process sometimes referred to as synaptic fusion. (TOS: "What Are Little Girls Made Of?") This was later offered to Uhura by Harry Mudd on the planet Mudd. (TOS: "I, Mudd")
"Practical immortality" was mentioned in "What Are Little Girls Made Of?", suggesting an eventual end. Uhura's model was stated to last only five hundred thousand years.
In 2369, Kobliad criminal Rao Vantika utilized a process akin to synaptic pattern displacement in order to transfer his consciousness to Doctor Julian Bashir. Repeated use of this process could be a practical form of immortality. (DS9: "The Passenger")
As of 2373 Tieran had existed for over two hundred years by transferring his mind to host bodies via a sophisticated neural implant. (VOY: "Warlord")
Leonard McCoy considered the advanced age Spock (and other Vulcans) would achieve to be close to immortality. (Star Trek V: The Final Frontier)
Several other species had long life spans, which could be considered as being close to immortality.
Trelane poetically stated that a kiss from Yeoman Teresa Ross would make him immortal. (TOS: "The Squire of Gothos")
Work related to Sherlock Holmes was considered immortal by Jean-Luc Picard. (TNG: "Lonely Among Us")
According to Beverly Crusher, Starfleet captains always acted as if they were immortal. (TNG: "The Battle")
It was hypothesized by Noonian Soong that procreation was done in part to give humanoids a sense of immortality. (TNG: "Brothers")
Onaya claimed her inspirations were beneficial. Although the ones she inspired died young, their works became "immortal." (DS9: "The Muse")
Lewis Zimmerman believed that by placing Julian Bashir's image on the Long-term Medical Holographic program, it would be a chance for Bashir to have "immortality." (DS9: "Doctor Bashir, I Presume")
Benjamin Sisko and Weyoun agreed that Vorta cloning was basically immortality, of a sort. (DS9: "Ties of Blood and Water")
Although the Borg discarded drones once they were irreparably damaged, their memories continued to exist within the Borg Collective. Seven of Nine compared this to the Human concept of immortality, and it helped relieve any fear of death. (VOY: "Mortal Coil")