Immortality

Immortality is the ability to live forever or avoid permanent death, either due to inherent abilities or use of external methods to prolong one's life indefinitely. Various life states throughout The Sims (series) possess this ability, and multiple methods exist to achieve immortality.

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Immortal life states

There are many life states throughout the series which are inherently immortal, and possess the capability to live forever unless killed through unnatural means.

The Sims

Ghosts cannot be killed due to already being dead, although they can be revived and killed again.

The Sims 2

Ghosts are immortal in essentially the same way as their counterparts from The Sims.
In The Sims 2: University, Zombies are unable to age past the Adult life stage, and therefore possess the potential to "live" forever.
In The Sims 2: Nightlife, Vampires, much like zombies, cannot age, although they can be killed by other means, notably their vulnerability to sunlight.
In The Sims 2: Open for Business, Servos cannot age, and can only be killed by a falling satellite, being eaten by the Cowplant, or running with scissors.

The Sims 3

Much like in the previous two Sims games, Ghosts are still essentially immortal, although they can be permanently removed from the game if their tombstone is deleted or they are made playable and choose to Return to Netherworld.
In The Sims 3: Late Night, while still possessing greatly augmented lifespans, vampires are no longer immortal and can die of old age, although if The Sims 3: Supernatural is installed they can acquire the Immortal lifetime reward to make them completely ageless and immune to the sunlight.
In The Sims 3: Into the Future, Plumbots, much like Servos, cannot die of old age and can only be killed by falling meteor, although some store items such as the Cowplant or the Door of Life And Death are capable of killing Plumbots.

The Sims 4

Ghosts remain immortal, but can be killed if they are revived first.

Methods of becoming immortal

The Sims 2

Sims with a high enough Aspiration Meter can drink the Elixir of Life and continously purchase resupplies of the elixir when they run out, effectively rendering themselves ageless. However, this method not only requires the Sim to maintain a high Aspiration Meter, but also requires the Sim or another Sim in the household to be able to afford replacement elixirs each time one runs out.
Similarly to the above method, if The Sims 2: University is installed, a Sim can use the elixir produced by the Cowplant to achieve the same effect. This has the benefit of not requiring a certain aspiration level but comes with the drawback of requiring other Sims to be sacrificed to the Cowplant on a regular basis.
If The Sims 2: University is installed, Sims can be brought back to life through use of the Resurrect-O-Nomitron by another Sim. Since this process can be repeated an indefinite number of times, any household can avoid permanent death using this method. A deceased Sim can also be revived as a zombie to prevent them from aging, although it is questionable if an immortal Sim would prefer such an existence.
If The Sims 2: Nightlife is installed, becoming a Vampire will make a Sim ageless but extremely vulnerable to sunlight.

The Sims 3

Ambrosia can be used as an effective means of prolonging a Sim's life, as it's consumption will reset a Sim to the beginning of their life stage and add eight days to their lifespan. However, preparing Ambrosia requires rare ingredients that generally necessitate a high skill level to obtain, as well as the $12,000 recipe and a level 10 Cooking skill.
A Death Flower can be kept in a Sim's inventory as a means to prevent permanent death. When a Sim carrying a Death Flower dies, the flower will be presented to the Grim Reaper, who will excitedly accept it and restore the dead Sim to life. This applies no matter how the Sim died, and a sufficient supply can render a Sim completely immortal.
Unlucky Sims are unable to die accidentally, as the Grim Reaper finds their misfortune too amusing and will simply revive them, providing a partial form of immortality.
If The Sims 3: Generations is installed, Sims with a sufficiently high Logic skill are capable of creating a Young Again potion, which will restore a Sim to the beginning of their Young Adult life stage. The potion can also be purchased as a lifetime reward.
The above expansion pack also includes an Age Freeze potion as a lifetime reward. Drinking the potion will render the Sim ageless, however if the Sim dies accidentally and is revived the effects of the potion will disappear and the Sim will age normally.
The Door of Life And Death can be used by any Sim to revive deceased Sims continously, although this method will reincarnate the Sim with a randomized appearance and gender, and sever the Sim's previous relationships as well as a portion of their skills. This method also requires that the reincarnated Sim be raised from babyhood back to adulthood, and may seem undesirable to some players.

The Sims 4

Ambrosia functions much like in the previous game but requires more skills to prepare, and can be used to prolong a Sim's life indefinitely.
A Potion of Youth can be purchased as an Aspiration reward, and can be used much like Ambrosia.
The Cowplant returns in this game, and unlike in the previous games, can be grown from a seed. It's elixir, gained from eating Sims, can be used to extend a Sim's lifespan.
The Book of Life can be used by a Sim to revive another Sim they are acquainted with, providing a potential means of immortality as long as there is a Sim to revive the one who dies.

Trivia

Most service NPCs, like firemen and the postal worker, cannot be killed unless added to the household through console commands.
The Grim Reaper is believed to be immortal, although it is difficult to tell as a new Grim Reaper could be generated by the game if the first one dies.
Despite being undead, Bonehilda is not immortal and can be killed.