TITLE: Non-commutative duality I: Which C*-algebras are (isomorphic to a) convolution algebra?
QUESTION [14 upvotes]: Many interesting C*-algebras can be realized as convolution algebras over a groupoid, an idea introduced in 1980 by Jean Renault (this entry in nLab provides plenty of context to the general approach of attaching an algebra to a groupoid).
Perhaps due to my incompetence in  this formidable field, I was not able to identify some results in the way of characterizing which C*-algebras are in fact convolution algebras.
So, here is my question:

Is there a duality theorem stating that the sub-category of C*-algebras satisfying (add here some properties) is dual to the category of locally compact groupoids ?

ADDENDUM: I have changed the title to make clear what I said in my comments below: though this is a very specific question (and by no means trivial, see the excellent responses below), it is just a tassel toward a broad goal, namely get some kind of duality theory for non commutative spaces. I was (happily) surprised not only with the answers I have received, but also by the views and the likes this question has gotten. That means that numerous folks share this interest. There will be further questions under the same rubric, and I hope they will attract other members of MO from a broad variety of backgrounds (operator theory, higher category theory, mathematical physicists, and generally researchers  with a passion for duality theory)

REPLY [3 votes]: There are already excellent answer so I hope my small remarks here will not seem to be too trivial. They concern the geometric origin, in a way, of the Fourier morphism mentioned by Simon Henry in a very specific contect.
When quantizing Poisson manifolds into NC $C^*$-algebras on possible way is through what is called groupoid quantization. The procedure is, here, to build the symplectic groupoid integrating the original Poisson manifold (thus a very specific kind of Lie groupoid) and then performing geometric quantization on this symplectic manifold keeping in mind, in a way, its groupoid structure.
The relevant part is the choice of a polarization compatible with the groupoid structure: what is called a multiplicative polarization. Such polarizations induce a groupoid fibration from the original symplectic groupoid to a quotient groupoid (which is not Lie but only topological) and a cocycle on this quotient groupoid coming from the symplectic structure. The (twisted by the cocycle) groupoid $C^*$ algebra thus resulting is the outcome of the groupoid quantization procedure.
Different choice of polarizations may result in quotient groupoids much differing one from the other and in the quotient cocylce being trivial or not. The typical example is an invariant symplectic structure $\omega_\theta$ ($\theta\not\in \mathbb Q$) on the torus which, depending on the choice of polarization, may be either quantized by the irrational rotation algebra (groupoid $\mathbb Z\ltimes_\theta \mathbb S^1$ with trivial cocycle) or by a trivial groupoid based on $\mathbb R^2$ (0-isotropy) but with non trivial cocycle depending on $\theta$. That the two are isomorphic may be considered a quantization does not depend on polarization type of result. The relation between the two is a partial Fourier transform applied to one of the two variables, in a way, and in general whenever the symplectic groupoid can be identified with a cotangent bundle as a manifold (which happens for a wide class of Poisson manifolds) different choices of polarization defines some Fourier-type relation between different (twisted) groupoid $C^*$-algebras.
One of the subtle points here is that in some sense groupoid $C^*$-algebras behave in a contravariant functorial way with respect to the base and in a covariant functorial way with respect to the isotropy so that all relations (like this choice of polarization) that result in some interchange of variables between base and isotropy have a quite complicated functorial description.

Eli Hawkins, A groupoid approach to quantization, J. Symplectic Geom. 6 Number 1 (2008) 61–125. Project Euclid, arXiv:math/0612363.