What is an Oper? Given a curve C, and a reductive group G, there is a moduli stack Loc_G(C), the stack of G-local systems.  I keep reading that there's a substack of "opers" but am having trouble locating a definition.  So what's an oper, and how should I think about them?
 A: Look at http://arxiv.org/abs/math/0501398 (Opers, Beilinson and Drinfeld, 1993/2005)
A: Edward Frenkel also has a number of papers which deal with opers (just look at his arxiv papers for example).  In particular, I'm fond of the paper he wrote with David Ben-Zvi http://arxiv.org/abs/math/9902068
I think this paper might be of particular interest to Charles given that I've seen him previously give links to the BNR paper.  This paper by Frenkel and Ben-Zvi relates spectral curves to opers and also gives some nice historical background on how each is related to solving certain kinds of differential equations (which is the kind of thing I'm glad to know is there, even if I don't study it that way).
A: For reviews, one can try the articles of Frenkel and Teschner.
In the physics context, a fairly recent article which uses opers is this one, where Gaiotto and Witten define an oper for $G = \text{SU}(2)$ to be a flat rank two complex bundle $E$ over a Riemann surface $C$, with structure group $SL(2, \mathbb{C})$, together with a holomorphic line sub-bundle $L$ in $E$ such that $L$ is nowhere invariant under parallel transport by the connection $\mathcal{D}_z$ on $E$.
An oper which is endowed with a covariantly constant reduction of its structure group to a Borel subgroup (ie. the group of upper triangular matrices) is called a Miura oper.  This notion is also described in detail in the above article by Frenkel.
