Do all 3D TQFTs come from Reshetikhin-Turaev? The Reshetikhin-Turaev construction take as input a Modular Tensor Category (MTC) and spits out a 3D TQFT. I've been told that the other main construction of 3D TQFTs, the Turaev-Viro State sum construction, factors through the RT construction in the sense that for each such TQFT Z there exists a MTC M such that the RT construction of applied to M reproduces Z. Is this true for all known 3D TQFTs? Does anyone know any counter examples?
Edit: 
(1) I want to be flexible with what we call a TQFT, so anomalies are okay.
(2) There have been some good answers to the effect that more or less if I have an extended TQFT then it factors through the RT construction. But this is not really what I'm after. Are there any (non-extended) examples that people know about? Ones which might not come from the RT construction. Are all known 3D TQFTs extended TQFTs?
 A: I think that the answer is "yes" if by TQFT you mean one that extends all the way down to 1-manifolds.  The MTC is the thing assigned to a circle by this extended TQFT.
There's also lurking somewhere in here the issue of whether you mean 3d TQFT "with anomaly."  The 3d TQFTs coming from modular tensor categories via RT really have a little bit of 4d structure.
Edit: Removed a false sentence.  Also note that precise statements (and eventually a proof, once all parts are out) can be found in work of Bartlett-Douglas-Schommer-Pries-Vicary
A: Kevin's parenthetical about needing things to be sufficiently finite and semisimple suggests that thought of another way the answer is "no."  In particular, there are known non-semisimple TQFTs.  I know very little about these, but Alexis Virelizier was very into them.  His papers (for example, this one) discuss them and should have references to earlier work.
A: There must be lots of fermionic TQFTs behaving in a very different way compared to RT. One feature of these fermionic (based on Berezin integral) theories is that they are quite straightforwardly generalized to any dimensions, not just 3 - which, by the way, provides you with a powerful and intriguing means of studying 3-manifolds using 4-manifolds, etc. These new theories are related to Lie groups (without any obvious restrictions as for semisimplicity etc.) and their homogeneous spaces, see e.g. http://arxiv.org/abs/0907.3787 .
A: If you have a 3d TQFT, with no anomaly, and which goes down to points, and where things are sufficiently finite and semisimple, then I think you can show that it comes from a Turaev-Viro type construction on the 2-category Z(pt).
If you have a 3d TQFT, possibly with anomaly, which goes down to circles, and where things are sufficiently finite and semisimple, then I agree with Noah: Z(S^1) is a MTC and the RT construction on the MTC reproduces the TQFT.
Relating these two statements, TV(C) = RT(double(C)), where C is a 2-cat, double(C) is the Drinfeld double (or maybe center), TV is the Turaev-Viro construction, and RT is the Reshetikhin-Turaev construction.
A: If your TQFT does not extend to the circle, (EDIT: and you're willing to generalise to lax TQFTs,) then the answer is no. A reference is: https://arxiv.org/abs/1408.0668
The proof goes by an explicit generators and relations presentation of the 3d cobordism category in terms of surgeries and automorphisms of surfaces.
