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Is there a noun that precisely describes the emotion of missing someone/something?
There is longing, but I don't think that's quite right. You can feel longing for something without missing it. I know this word exists in another language (Hebrew - געגוע), but I'm not sure if there is an equivalent in English.
Besides previously-mentioned pining, yearning, and longing, consider wistfulness, "state or characteristic of being wistful", that is, being "full of yearning or longing". Other nouns include yen, "a strong feeling that you want to do something"; craving, "a strong desire; yearning"; thirst, in its figurative sense "a want and eager desire after anything; a craving or longing"; and nostalgia, in its senses "a longing for home or familiar surroundings" and "a bittersweet yearning for the things of the past".
My guess is that there is no such exact noun, but for some special purpose, you could use "absence" suitably modified, with the concomitant wordiness perhaps serving a rhetorical (e.g., humorous) purpose, e.g., saying "With my girlfriend away on the long business trip, I'm experiencing that sort of absence that makes the heart grow fonder." sounds more inviting of a sympathetic smile that the neutral / merely factual, "I really miss my girlfriend who is away on a long business trip."
Perhaps saudade (Portuguese) can be used. While not yet a common loan word, it is seen in English contexts sometimes. Wikipedia says this, and a whole lot more ;) :
It describes a deep emotional state of nostalgic or deeply melancholic longing for an absent something or someone that one loves.
Remembrance could be a kind of missing of (remembering of) someone or something without the element of longing for that person or thing. It is like nostalgia without necessarily yearning to return to the past.
I like jwpat7's suggestion of wistfulness, if the word you're looking for has an edge of sadness.
