Quotients of Abelian varieties by finite groups Let $A$ be an Abelian variety (over an algebraically closed field).  The group $\mathbb{Z}/2\mathbb{Z}$ acts on $A$ and the quotient is called the Kummer variety of $A$.  These are well studied and, as I understand it, their geometry is understood and a classical subject (especially for surfaces).
What happens when you take the quotient of an Abelian variety by other finite groups?  Is the geometry of the quotient understood in any sense (e.g., Kodaira dimension)?  Are the quotients somehow classified?  Is anything else known about them?
 A: The quotients of abelian surfaces (over $\mathbb C$) by finite groups are classified by Yoshihara. In particular he determines the possible Kodaira dimensions.
For instance, if the holonomy part of the group ( quotient by its maximal translation subgroup ) has cardinality greater than $24$ then he shows the quotient is rational.
The precise reference is

*

*Yoshihara, Hisao. Quotients of abelian surfaces. Publ. Res. Inst. Math. Sci. 31 (1995), no. 1, 135--143.

Unfortunately, I am not aware of any electronic version of this paper.
For higher dimensional abelian varieties I am not aware of any work studying the
finite quotients. But for 3-dimensional complex tori  there is for instance this paper by Birkenhake,  González-Aguilera, and Lange which classifies the possible finite subgroups. The same
authors also have a paper dealing with finite subgroups of the  $3$-dimensional abelian varieties (over $\mathbb C$ if I remember correctly).

EDIT (May 18)
You may want to take a look at Complex crystallographic groups I and II. The authors study  compact quotients $X$ of $\mathbb C^n$ by discrete  subgroups $\Gamma \subset \rm{Aff}(\mathbb C^n)$.
In dimension two they obtain classification results for the pairs  $(X, \Gamma)$ assuming
$\Gamma$ (more precisely its holonomy part) is generated by reflections (paper I); or $X$ is rational (paper II).
A: There is a classification of finite groups acting freely on abelian threefolds, leading to quotients which are Calabi-Yau threefolds, in Oguiso, Sakurai, Calabi-Yau threefolds of quotient type, arXiv:math/9909175. 
A: You may be interested in a series of papers I wrote with Giancarlo Lucchini Arteche:
Smooth quotients of abelian varieties by finite groups (A characterization of smooth quotients of abelian varieties of dimension $\geq3$ by finite groups that fix the origin)
Smooth quotients of principally polarized abelian varieties (A characterization of smooth quotients of abelian varieties by finite groups that fix the origin as well as a principal polarization; this gives a moduli-theoretic spin on the problem)
Smooth quotients of complex tori by finite groups (with an appendix by Stephen Griffeth) (Here we provide a "bridge" between the case of smooth quotients where the group fixes the origin and the case of a free action. Essentially general smooth quotients of complex tori by finite groups are fibrations in products of projective spaces over an étale quotient of a complex torus)
There is a lot to be done in the case of étale quotients, and as the other posts say, much work has been done in this case by Uchida and Yoshihara, Lange, Catanese and Demleitner.
A: If you are still interested at the moment there is a complete classification in 3 and 4 dimension when the group acts freely on the abelian varieties (they are called hyperelliptic varieties). Some authors are Catanese and Demleitner: if you search their papers you can find others (and better) references.
Maybe you are also interested in the so called generalized kummer.
