To date, 8 of the problems have been resolved, meaning the problem was solved or a proposed proposition was proven or disproven. Another 9 have been partially resolved, meaning there exists some controversy as to whether the proposed solution fully resolves the question. Six problems remain unsolved.
A breakdown of the problems' status is given here: Hilbert's problems - Wikipedia.
Problem 2 was written about in a prior post about Kurt Gödel's incompleteness theorem.
Problem 18 sought a proof for the densest sphere packing which wasn't resolved until 1998. Also see this blog's post on sphere packing in higher dimensions.
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