Mike's List of the Most Important Mathematical, Scientific, and Engineering Challenges for the Twenty-First Century.

I occasionally dabble as a futurist; when I do this, I try to take a broad perspective on the long-term future of our civilization and how science & technology can/will/might affect it.  Below is my own compilation of what I think will be the biggest, most important, and most fundamental mathematical, scientific, and engineering challenges that may impact the future of human society over the next century or so.  Along with the engineering challenges, I recommend social cautions.

If you have any suggestions for additions to this list, please let me know.

Mathematical:

1. (Very hard, perhaps unsolvable.)  Prove whether or not P=NP, or at least prove whether or not the question is independent of the axioms of (conventional models of) computation.  If the proof is that P=NP, provide an effective deterministic algorithm for simulating nondeterministic machines.  This would make all other mathematical questions much easier to answer, as it would enable automatic generation of proofs in time  polynomial in the proof length.  (This is only useful if the constants are small enough!)
2. (Very hard.)  Prove whether or not quantum computation can be simulated in polynomial or at least sub-exponential time by classical computation; if it can, provide such an algorithm.  This is equivalent to proving whether quantum theory implies an exponentially more complex underlying structure for our universe than the classical appearance that we perceive.  If a polynomial simulation exists, it makes it much easier to compute the consequences of quantum theories, without requiring a quantum computer.
3. (Assuming the answer to #2 is negative) Prove whether or not a scalable quantum computer is theoretically feasible, given reasonable constraints on how the technological parameters (precision, cost, etc.) must scale with increasing computational complexity.  This problem is not too hard, and theorists are presently making rapid progress on it. 
4. (Assuming the answers to 1 and 2 are negative, and that 3 is positive) (Hard.) Prove whether or not quantum computers are capable of solving NP-complete problems in polynomial time.  If positive, the answer could revolutionize mathematics (see #1).
5. (Given a final theory of physics, from #9) (Very hard.)  Determine once and for all whether any form of time travel is theoretically possible given known physics.  (A sufficiently efficient and precise interactive computational simulation of the earlier history of the universe might comprise one virtual solution.)

Scientific:

6. (Assuming #11 has been done.)  Use quantum computers to simulate quantum physics, and to predict experimental consequences of quantum theories that are currently intractable to analyze.
7. (Perhaps with help from #6) Via theory and experiment, narrow down the possibilities for a unified theory of general relativity and quantum mechanics to a single, well-validated model.  Do whatever else is necessary to achieve a final theory of physics (accomodate any surprising new phenomena that are discovered, etc.).
8. Decipher the detailed functionality of the brain, and understand the phenomenon of intelligence.  May require nanotechnological tools from #12.
9. Determine the density of life, and intelligent life, in the universe.
10. Determine the ultimate cosmological fate of the universe, & whether we can possibly affect it.

Engineering:

11. (If #3 turns out positive) Build a working, scalable, and cost-effective quantum computer.  Be cautious of possible societal dangers during the transition from classical to quantum cryptography.
12. (Perhaps with help from #6) Create an advanced nanotechnology that offers complete control over atomic placement, sophisticated molecular mechanical & electronic designs, and self-replication.  Develop its applications in all areas; esp., mass-manufacturing, medicine, ecosystem management, spaceflight, and computation.  Be very cautious of possible societal dangers from malicious use of the technology.
13. Create true artificial intelligence; also enable the mirroring of human minds to artificial forms.  (Probably depends on #8.)  Be very cautious of possible societal dangers from intelligent but inhuman minds run amok.
14. Begin colonizing the universe.  Be very cautious of possible societal dangers from other intelligent life-forms, if we determine from #9 that there may be any.