Orbifold with discrete torsion I'm trying to understand some of the early works of Vafa and Witten [1-3]. The way I look at orbifolds is they are the quotient space $M/G$. This is simply a quotient manifold when the action of $G$ on $M$ does not have any fixed points, but if it does then this is not a manifold but an orbifold. I don't see how exactly "orbifolding" takes care of the singularities? 
And what exactly is an orbifold with discrete torsion ? Despite many research I could not find any source where orbifolding or orbifolds with discrete torsion is introduced and discussed in a physics point of view. Any reference? 
Thanks! 
[1] Vafa, Witten; On Orbifolds with Discrete Torsion 
[2] Vafa; Modular invariance and discrete torsion on orbifolds
[3] Dixon, Harvey, Vafa, Witten; Strings on orbifolds
 A: What came to be called "discrete torsion" is simply the data that makes the B-field gerbe be equivariant over the orbifold. This was clarified by Eric Sharpe, see the references here:
Eric Sharpe,
Discrete Torsion and Gerbes I (arXiv:hep-th/9909108)
Discrete Torsion and Gerbes II (arXiv:hep-th/9909120)
Discrete Torsion, Quotient Stacks, and String Orbifolds (arXiv:math/0110156)
A: I can only answer the mathematical part of your question (or make a stab at it). We could say that by describing a space as an orbifold, the singularities are taken care of by somehow declaring them to be under control.
Where a manifold is a topological space that may be very complicated, but locally looks very nice, namely like $\mathbb R^n$, an orbifold locally still looks very nice, though slightly less so (or rather, slightly more general), namely like the orbit space of $\mathbb R^n$ under the action of a finite group. 
In reality this local quotient may still look like $\mathbb R^n$, and the local, linear group action is part of the orbifold atlas, so actually it is an additional structure on the space.
An intermediate class of spaces is that of manifolds with boundary. This is a space that locally looks like a Euclidean half-space.
Rather than saying that an orbifold is a space of the form $M/G$, I would say that an example (the main example) of an orbifold is a quotient space $M/G$ (where $M$ is a manifold and the group action is sufficiently good).
