As you know, the Hilbert sixth problem was to axiomatize physics. According to the Wikipedia article, there is some partial succes in this field. For example, Classical mechanics, I believe, can be treated now as an axiomatized discipline since it is properly formalized in the modern theories of Lagrangian and Hamiltonian mechanics (and as a corollary, one can consider it as an extension by definition of the theory of real numbers, which in its turn is an extension by definition of the axiomatic set theory -- and eventually all these theories can be treated as first order theories).
I am not a specialist, but the Theory of relativity seems to be formalized as well in terms of differential geometry (is that correct?). On the other hand, as we understood not long ago, the Quantum mechanics is not axiomatized. I often discuss this with my colleagues and students, so I believe this is an important question, it deserves specification and clarification, the Wikipedia article is too short, it must be detailed. I believe there are people here who can explain clearly
which parts of physics are axiomatized now, and which are not.
For the educational purposes (and I am sure, this will be interesting for specialists as well) it would be good to have a list of these disciplines. In the discussions at the web, for example, in the discussion at MathStackExchange, I don't see this list. If it is possible, I ask specialists to share their knowledge. There is no necessity to list all the disciplines in one answer, each answer can be devoted to one discipline, I only ask people to give the necessary references and to provide some elementary explanations (of course, detailed answers are better) so that non-specialists (and students) could understand.
EDIT. From Ben Crowell's answer I deduce that there is some discrepancy in understanding of what is meant by the 6th Hilbert problem: some people interpret it as a suggestion to construct a first order theory of a given physical discipline (this means, to build a "completely new theory", formally independent from the other mathematics, with its own language of logic -- like in the modern set theories, ZF, NBG, MK, which are the standard examples of the first order theories), while others interpret this wider, as a possibility to give any system of axioms, not necessarily first order (this can be understood as building an extension by definition of another, already constructed first order theory -- and the standard examples are Hilbert's axioms of Euclidean geometry, Kolmogorov's axioms of probability theory, etc.). As to me I don't see serious reasons to restrict ourselves on first order theories: the "axioms as definitions" are quite satisfactory, moreover, I would say they are preferable, since they do not deflet attention from the essential part of the problem (to the details with the new language of logic, etc.). Actually, if we look at the problem from this "practical" point of view, I do not see the difference between "axiomatization" and "formalization", that is why I referred to the books by V.I.Arnold and A.L.Besse from the very beginning.