Some of Hilbert's Problems:

Problem Resolved?
1. The Continuum hypothesis Partially Independent of the axioms of arithmetic
2. The axioms of arithmetic are consistent Partially See Godel's Theorem
... ... ...
7. Is $a^b$ a transcendental number, for algebraic numbers $a\ne{}0,1$ and algebraic b ? Yes See Gelfond-Schneider theorem
8. The Riemann hypothesis No One of the Millennium problems

David Hilbert at the beginning of the 20th Century formed a list of twenty-three Mathematical problems which were unsolved at the time as a challenge to Mathematicians. They have been very influential in the development of Mathematics since that time.

Many have been resolved but some including the Riemann Hypothesis await a decision.

The complete list and their status can be found on the WikiPedia page:

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