Reincarnation is the idea that a person’s eternal soul is reborn numerous times in different bodies. In a religious sense, reincarnation tends to occur as a purifying process: through each life cycle, you either learn important lessons, cleansing your soul and becoming closer to heaven, or rack up karmic negativity, setting your eternal self back in the queue and, depending on the belief system, guaranteeing a shittier vessel for yourself in your next life cycle. The monotheistic Big Three—Judaism, Islam, Christianity) don’t generally hold with this concept, but it is an important part of many Eastern religions, such as Buddhism and Hinduism.
Reincarnation is also a time-tested trope in fiction, and I find it interesting that in most cases, it’s a hard and fast aspect of the fictional universe rather than a religious belief that some follow and some don’t. It’s just given that reincarnation exists.
Saiyuki sort of gets a pass on that, I guess, since a series entirely based on the Journey to the West story, which details the founding myth of Mahayana Buddhism, is going to assume the belief system it’s referring to is real. It’s not like it’s about characters who are Buddhist and believe they have been reborn without proof. We see two different life cycles of three of the four main characters of Saiyuki (Sanzo, Gojyo, and Hakkai): one of their past lives, where they were gods who died defending Goku in Heaven, and then their current life cycle, where they are mortals on a journey to the far West. The reincarnation in this story is true to the Buddhist concept of reincarnation, but it doesn’t do much for the storyline except to reinforce the idea that the characters are no-good rebels, regardless of when or where you find them.
Avatar: The Last Airbender and Legend of Korra’s titular Avatars gain their powers through an unbroken line of reincarnation. This is yet another example of a world where reincarnation isn’t a belief, it’s an observable reality. The Avatar is reborn in an unending cycle of one of the four races of its universe (water, fire, earth, and air tribes), and can, through meditation, reach the Avatar state, a mental state where he or she can converse with his or her past selves. This series is strongly based in Eastern traditions, from the world-building to the people and history, so it’s unsurprising that reincarnation would play a large role in the show. It functions here mostly as a plot forwarding device—entering the Avatar state is a pretty standard way to get out of a jam, as it allows Aang or Korra to communicate with past Avatars and allows said Avatars to exposition-bomb. The difference between this sort of reincarnation and, say, Buddhist concepts of reincarnation is that the Avatars aren’t working toward an eventual spiritual goal like heaven or nirvana—there’s no ongoing purification or religious journey with a higher purpose. Each Avatar serves the people in his or her lifetime and then dies and is reborn to do the same thing again.
The Wheel of Time series is an interesting example of reincarnation in fiction because its worldbuilding is pretty solidly based in the pseudo-European fantasy tradition. Although the books span several countries with unique cultures, the heroes still tend to be the ones from the sort-of-like-medieval-England places, so it’s a little different to have reincarnation play such a large role in a series not really based around an Asian mythological tradition. 
Wheel of Time is another series where the cycle of rebirth is just a given thing accepted by everyone—furthermore, it’s a major plot element, as a large conflict throughout the series is whether Rand al’Thor, the protagonist, is actually the Dragon Reborn (‘the Dragon’ being the title given to Lews Therin Telamon, a legendary leader of a past Age). I find the use of reincarnation in this series very interesting, especially as it affects different characters’ relationships. Certain characters who are, for one reason or another very long-lived, will base their interactions with Rand on their interactions with his past-Age counterpart; Rand has visions of his past life, and even is able to communicate with Lews Therin, an act which threatens Rand’s sanity and grip on reality. The high fantasy setting of this story leaves off any really religious ideas—priests and scriptures are exchanged for sorceresses and prophecy—so although the Wheel of Time is basically literally the wheel of samsara, the cycle of death and rebirth in Hinduism and Buddhism, the two traditions don’t have much other influence on the story.
Reincarnation is an interesting concept in fiction, but I think it’s hard to pull off in a universe where it isn’t an accepted reality for everyone. In a world where a cycle of death and rebirth was part of only some religions and not others, I can’t imagine it would be able to play as big a role. It would be hard to get people to believe that Rand (and not someone else) was the Dragon Reborn if they have trouble accepting that Lews Therin could even be reborn in the first place. I do think, though, that it can be very interesting in fiction where it’s used correctly, offering a lot in terms of character development and allowing for grand, historical plots and destiny-driven storylines.
What are some other good examples of reincarnation in pop culture? Let me know in the comments!
Lady Geek Girl and Friends 
You have to read a book called The Years of Rice and Salt. It’s an alternate history book which re-writes history as if the black death had completely wiped-out Europe, leaving the Asian nations to conquer the world. It tracks a few characters through at least a dozen reincarnations, over hundreds of years. In almost all of these reincarnations the characters are not aware of their greater connection, and in fact for many of the reincarnations they affiliate which non-reincarnating religions. They even change gender many times and meet each other at different ages. In one life they may both be young men; in the next a mother and daughter. This makes the book very interesting.
Also: I think the nature of the Avatar’s reincarnation (no ongoing journey of self improvement) equates with the Dalai-Lama.