Large eyes emphasise the youthfulness and cuteness of a character. They also make it easier to convey complex and nuanced emotions. Eyes are an important tool for emotional communication for humans; the difference between Japanese and occidental smileys (which focus the information on eye and mouth shapes, respectively) would suggest that the importance of eyes is even higher in East Asia.
Exaggeratedly large eyes are one of the most famous and recognisable elements of anime and manga. Large eyes are ubiquitous in shonen and especially shoujo, which also happen to be the two best known demographics outside of Japan. Chibi style, again common in manga and anime aimed at kids and teens, exaggerates the eye size even further. Also it has to add that older shonen (and some newer ones) avert big anime eyes, instead the eyes tend to be either narrower or realistic.
With "anime eyes" not only the size increases, but also the shape is exaggerated (and usually gives a hint whether the character is generally active or passive), the iris and the pupil grow in proportion and change shape to compliment the general eye shape, and especially in the romantic genre, the reflections are greatly exaggerated. Horizontal lines above the eyes represent the folds of the upper lid and are not necessarily realistic for Asian characters, but serve as a visual anchor that makes for a lightning-fast distinction between a narrow eye and a half-closed eye. The range and flexibility of the eyebrows is also exaggerated.
Thus big eyes are very very popular for characters intended to be cute and endearing, and in genres which rely greatly on emotion and melodrama. Even in works that employ more realistic proportions use eyes as a major tool. Only in the gag genre and some more experimental styles are eyes downplayed.
On the other hand, characters with small and possibly oversimplified eyes are usually intended to be incredibly ugly, and are used more as tools for gags rather than fully-fledged characters. Characters with no eyes at all are the faceless mass, part of the background, moving standies, or perhaps intended to be as emotionally distant from the other characters and the reader as possible.
Interestingly enough, we actually partially have western animation to thank for this. The trend of big eyes in Japanese Media was started by Osamu Tezuka, considerated by many as The Father/Godfather/God of Manga, who was influenced by western cartoons such as Classic Disney Shorts, Fleischer cartoons like Betty Boop, and films like Bambi to get this kind of eyes in his creations, notably Astro Boy. His works became so important for the industry (first manga and later anime) that the concept of "anime eyes" was widely accepted and much of the works use this kind of eyes until today.
"Anime Eyes" is an Omnipresent Trope in Japan, so the list of works would be huge and widely common, so this trope is about Western or non-Japanese works that use this as a resource intended to be "anime eyes". In the case of works from Japan, only should be added if it's the case of Lampshading, Subversions and Aversions.
Compare Puppy-Dog Eyes, Anime Hair and Disneyfication or Cartoony Eyes (a Western version). Strongly related to Windows to the Soul, as well with Good Eyes, Evil Eyes. See also Mukokuseki, which applies this to everyone regardless of ethnicity, Animesque and OEL Manga, in which Western and non-Japanese creators draw their creations with big anime eyes as a way to emulate and/or make tribute to Japanese Media.
- If well it's a trope present in almost all manga and anime works, there're creators that try to avoid the "big anime eyes" making them more small and realistic. Some notable examples are the works of Hayao Miyazaki, Katsuhiro Otomo and Takehiko Inoue, just to name some big names on industry.
- Though it's a Studio Gainax production, and Panty & Stocking with Garterbelt mostly adheres to this trope, it's pointedly averted in the episode "Vomiting Point." As Garterbelt succinctly describes it: "In this ghetto-shit place [Little Tokyo], ghetto-shit people seek ghetto-shit happiness as they live out their ghetto-shit lives." Reflective of their miserable existence, the people are drawn with tiny, mismatched eyes◊, bordering on gonk.
- Bleach anime The Lost Agent/Fullbringer/XCution arc. The Fullbringer/XCution character Riruka Dokugamine Lampshades this in one episode when she says that her eyes dry out easily because they're so large.
- Franken Fran: One strip revolves around a girl who asks Fran for surgery to look like a chibi manga character to get her crush's attention (a supporter of the "2D girls are superior to real life" debate). She keeps coming back to Fran for more modifications, until she finally has the opportunity to sleep with him, at which point we see the extent of the changes: her eyes are half the size of her head, her mouth is shrunken, her nose is gone and her hair fell out◊: she looks like one of The Greys. Unsurprisingly, the guy swears afterwards that he experienced alien contact.
- Goodnight Punpun has a relatively realistic art-style devoid of the usual manga character design cliches. There is at least one Take That! at big-eyed manga characters.
- Lampshaded in an issue of the The Simpsons comic book where Comic Book Guy lived his life as a Japanese cartoon, and did so with the help of some extra-large contact lenses.
- Referenced in the web round-robin short story Lungfish Alpha features a future in which people can make quite literally whatever physical modifications to their bodies that they want. The main character, Itsuko, was born to a pair of anime Otaku modifications.
They'd geneered their daughter to resemble the characters they loved: ridiculously colored hair, extended legs and arms, a tiny mouth and eyes that took up 70 percent of the face.
- The Doomy Adventures Of Irken Doominess: All characters created by Jir will automatically have these sort of eyes.
- Invoked on Robots. As Fender is taking photos of Rodney, he asks for "big anime eyes".
- The Alita: Battle Angel adaptation of Yukito Kishiro's creation has Rosa Salazar as Alita. Apart of the Race Lift polemic of their characters, Alita specially is in the eye of the storm, because of the anime eyes made by CGI as seen in the first trailer. Director Robert Rodriguez explains the reason why he made Alita with anime eyes in this IGN interview.
- The 2014 biopic Big Eyes is about the life of Margaret Keane, a famous American artist that painted women as well children and animals with big and expressive eyes before anime was known in Western.
- In the fourth book in the Uglies series by Scott Westerfeld, there's a clique called "manga-heads", who get surgery to make their eyes appear larger and who have crazy hairstyles in order to look like they're from manga.
- Big Eyes, Small Mouth lampshades this in its title.
- Phineas and Ferb: When Phineas and Ferb visit Tokyo, they briefly Art Shift into anime-style, in particular, their eyes change to be huge and sparkly.
- Despite taking place in the equivalent of The Roaring '20s, Ikki from The Legend of Korra draws in a manner akin to 1990s shoujo manga, complete with big eyes.
- The Powerpuff Girls (2016) takes the franchise's animesque styling to its natural extreme in the Bliss multi-episode special. During it, there's an Art Shift sequence complete with a woman having huge, sparkling anime eyes.
- Here's what Roger Ebert (probably the most famous film critic in the United States) has to say about anime eyes.
- If well it's almost impossible to get the eye size for normal humans, there's a way to get the optical effect of "anime eyes" via makeup as seen in the main image, having many tutorials about how to make this on YouTube and Tumblr.
- A famous (and non-anime related) example is The '50s American artist Margaret Keane, who was famous for drawing paintings with big eyes, and mainly paints women, children and animals in oil or mixed media. Her story, as well the case of the trial of her ex-husband Stealing the Credit of her work, was shown in the Tim Burton's 2014 biopic Big Eyes with Amy Adams as Margaret Keane.