TITLE: Why is elimination of imaginaries important?
QUESTION [7 upvotes]: Some theories eliminate imaginaries and some times it takes a long to prove that, so why is this property important and what consequences does it have?
(for example the thoeory of Algebraically closed valued fields)

REPLY [6 votes]: I'm a beginner and this is unfair if I try to add something to Alex Kruckman's perfect answer. But I've seen an interesting tip as an idea for the importance of this concept in general. Roughly speaking, suppose we are in the category of sets, one can hold a bijective correspondence between surjective functions on given set, say $X$, and equivalence relations on $X$. Now if we go into the category of definable sets of a first-order theory, let say $Def(T)$, this correspondence generally fails, because when $E$ is a definable equivalence relation, maybe there doesn't exist corresponding definable surjective function $f_{E}$.