One of them considers the physical sciences, and mentions the kinetic theory of gases and the "laws of the motion of rigid bodies" as subjects which merit an axiomatic treatment.  
One of them which involved Fuchsian systems and monodromy groups was addressed by Josip Premelj, but his work was shown to be flawed by Andrei Bolibruch.  Another was addressed using the "p-adic" system developed by Helmut Hasse, while Montgomery and Zippin extended the work on one of them by Andrew Gleason.  One of them considers the physical sciences, and mentions the kinetic theory of gases and the "laws of the motion of rigid bodies" as subjects which merit an axiomatic treatment.  The first to be addressed was considered by Max Dehn, who showed that a regular tetrahedron and a cube of the same volume cannot be divided into identical piles of pieces.  Some of them involved the "most general law of reciprocity in any field," "building up space from congruent polyhedra," and the extension of Kronecker's theorem on Abelian fields.  The first of them asked about Cantor's problem of the cardinal number of the continuum.  FTP, name this program for research which was announced at a lecture in 1900, in which a German professor suggested that his colleagues work on 23 questions.
One of them which involved Fuchsian systems and monodromy groups was addressed by Josip Premelj, but his work was shown to be flawed by Andrei Bolibruch.  
Hilbert's problems or the Mathematical Problems of David Hilbert