there are MANY fundamental open problems
the following are I think truly fundamental querstions, (not just 'ow a problem we haven't solved, but that can't take long')
1) Why is space (at low energies for the string theorists..) four dimensional? and why is it (1+3) (so one timelike, and three spacelike dimensions) what are -on a fundamental level- the reasons why time behaves differnently then space?
2) What is dark energy? What is dark matter?
3) how can we unite general relativity and quantum field theory?
4) Why does Nature appear to follow some logic, given by conservation laws and symmetries (these two are strongly related)
5) what fixes natures constants? are they realy constant?
6) what is (quantum) information?
7) is mathematics the correct language to describe nature?
8) why are there three leptonic families? (see
http://particleadventure.org/particl...ess/chart.html ) (this one is probably related to (5))
good luck!
Bo