Hilbert's 23 problems are:
Problem 1  solved  The continuum hypothesis 
Problem 2  solved  Are the axioms of arithmetic consistent? 
Problem 3  solved  Can two tetrahedra be proved to have equal volume (under certain assumptions)? 
Problem 4[?]  too vague  Construct all metrics where lines are geodesics 
Problem 5  solved  Are continuous groups automatically differential groups? 
Problem 6[?]  open  Axiomatize all of physics 
Problem 7[?]  partially solved  Is a^{b} transcendental, for algebraic a ≠ 0,1 and irrational b? 
Problem 8[?]  open  The Riemann hypothesis and Goldbach's conjecture 
Problem 9[?]  solved  Find most general law of reciprocity in any algebraic number field 
Problem 10  solved  Determination of the solvability of a diophantine equation 
Problem 11[?]  solved  Quadratic forms with algebraic numerical coefficients 
Problem 12[?]  solved  Algebraic number field extensions 
Problem 13[?]  solved  Solve all 7th degree equations using functions of two arguments 
Problem 14[?]  solved  Proof of the finiteness of certain complete systems of functions 
Problem 15[?]  solved  Rigorous foundation of Schubert's enumerative calculus 
Problem 16[?]  open  Topology of algebraic curves and surfaces 
Problem 17[?]  solved  Expression of definite rational function as quotient of sums of squares 
Problem 18[?]  solved  Is there a nonregular, spacefilling polyhedron? What's the densest sphere packing? 
Problem 19[?]  solved  Are the solutions of Lagrangians always analytic? 
Problem 20[?]  solved  Do all variational problems with certain boundary conditions have solutions? 
Problem 21[?]  solved  Proof of the existence of linear differential equations having a prescribed monodromic group 
Problem 22[?]  solved  Uniformization of analytic relations by means of automorphic functions 
Problem 23[?]  solved  Further development of the calculus of variations 
According to Gray (see reference below), most of the problems have been solved. Some were not completely defined, but enough progress has been made to consider them "solved". He lists the fourth problem as too vague to say whether it has been solved. He also lists the 18th problem as "open" in his 2000 book, because the spherepacking problem was unsolved, but a solution to it has now been claimed (see reference below). Advances were made on problem 16 as recently as the 1990s, and progress continues. Problem 8 contains two famous problems, both of which remain unsolved. The first of them, the Riemann hypothesis, is one of the seven Millennium Prize Problems, which were intended to be the "Hilbert Problems" of the 21st century.
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