There's still unsolved problems - even everyday ones. The physics of bicycles (why it's easy to stay upright on one that is moving but not on one that is still) is not well understood. Just recently (like last year, I think) some guys finally explained why grapes emit plasma when you put them in a microwave. 

The moon is too lightweight (as in, the observed mass does not match up with the predicted mass). Last year the Pioneer anomaly was finally explained, though nobody at NASA was actually paid to work on it.