Curves of constant curvature on S^2 Most probably this is a well known question.
Consider $S^2$ with a Riemannian metric. I would like to ask what is known about the structure of the set of simple (without self-intersections) closed curves on it of constant  geodesic curvature. 
Here is a series of questions.
1) Is this  true that through each point of $S^2$ passes a simple closed curve of constant curvature? If not, can one estimate from below the proportion of the area of $S^2$ covered by such curves?
2) Is it true that for each value of curvature there are at least $2$ simple closed curves on $S^2$ of this curvature? Or maybe even more than $2$?
3) What can be said about the global structure of these curves on a generic $S^2$?
Taking the union of all such closed curves we could try to cook up from them a surface
(that maps naturally to $S^2$). Is something known about the topology of this surface?
Comments
1)  The theorem of Birkhoff states that each Riemannian $S^2$ contains at least three simple closed geodesics, as Joseph remarks below.
2) For a generic metric on $S^2$ the set of such curves  this set should be one dimensional. Indeed for each fixed value of curvature you can consider an analogue of the geodesic flow on the space of unite tangent vectors to $S^2$ and you expect that closed orbits will be isolated. 
ADDED. Is seems indeed that these are open (and I guess hard) questions. Macbeth gave a very nice reference, that tells in particular that similar questions were raised previously by Arnold, I copy the Macbeth's reference here, so it is visible to everyone: http://count.ucsc.edu/~ginzburg/ARNOLD/mag-post.pdf
Update. The following reference : http://arxiv.org/abs/0903.1128 gives a positive answer to question 2) for spheres of non-negative Gaussian curvature provided we consider not only simple curves on $S^2$ but also curves that bound immersed disks.
One more update. There is a new nice article  http://arxiv.org/abs/1105.1609 that provides some further results concerning question 2) for $S^2$ of positive curvature. This article also gives all necessary references from which one can conclude that 
question 2) was considered by Poincare in 1905, as it is written in the article of S.P. Novikov  http://iopscience.iop.org/0036-0279/37/5/R01/pdf/0036-0279_37_5_R01.pdf
 A: Matthias Schneider has a nice treatment of the problem:
http://arxiv.org/abs/0808.4038
Grossly speaking, generically the space if such curves is finite, and the number, counted algebraically, is equal to 2. If the ambient space is 1/4 pinched, then the curves are simple - they do not self intersect.
A: This is an update on new results (and old conjectures) on closed curves of constant curvature. I've just spotted a new paper on arxiv on immersed curves of constant curvature. The authors prove the existence of immersed closed curves of constant geodesic curvature
in an arbitrary Riemannian $2$-sphere for almost every prescribed curvature.
https://arxiv.org/pdf/2106.12374.pdf
There is a nice short survey in this paper about the history of the topic, on page 2. In particular, there is a reference to the following paper of Ketover and Liokumovich
https://arxiv.org/pdf/1810.09308.pdf
which attributes the following conjecture to Novikov:
Conjecture (Novikov). Every Riemannian two-sphere contains a smoothly embedded curve of curvature c for any $0 < c < \infty$.
The authors say that this conjecture is from Section 5 of the following Novikov's paper:
http://www.mi-ras.ru/~snovikov/74.pdf
I was not able to spot this conjecture in Novikov's paper immediately, but hopefully it's there. So it looks like part 2 of the original question is indeed an old open problem.
A: In a recent paper, Sun proved that
i)such curves concentrate around the critical point of the Gaussian curvature
ii) there exits a curve with constant geodesic curvature in every neighborhood of a non-degenerate critical of the Gaussian curvature 
My intuition is that we have 
ii') there exits a foliation of a neighborhood of non-degenerate critical of the Gaussian curvature foliated by curves with constant geodesic curvature and this foliation is unique
Since we have such a result for surface with constant mean curvature, see Ye91 and Ye96.
So this gives a picture of the asymptotic structure of this moduli space as a one dimensional manifold. However i guess that that the question of the global structure is quite open.
