Here are some related to science:

• Computation: Proof for an P not in NP theorem (is all the problems who are not able to compute in a deterministic polynomial (P) time belong to those who are computed in the non-deterministic polynomial (NP) time), a $1M question.
• Math: Proof for the last Ferma theorem, also a $1M question.
• Physics: Is there an extraterrestrial life?

We also have little understanding about how human brain works and complex decisions are made by the living beings as there are either no tools that are able to discover these nuances or the experiments raise ethical questions.

I also found there are some problems in math, particularly in topology, that no one has found the way how to solve them yet.

I am not including philosophical questions, such as those that are related to thought, personality, the universe, etc., as these are a matter of a definition.

Finally, there are many unanswered problems to explore about every individual person. Human is a very complex being and there is no free lunch way of understanding each individual the same way. So keep your best listening and analysing and try to make someone's life better :)

A probably highly topical question these days: Are computational word-sense disambiguation and computational word-sense induction possible?

hard consciousness

free will