The Times [UK] notes several areas in which modern science has made little or no headway in solving important questions.
Just what is dark matter and dark energy? This seems important in that they seem to account for 96 percent of the cosmos. We haven't a clue.
Why are the pioneer spacecraft drifting off course? Something is altering their trajectory and we don't know what it is.
Are physical constants constant? It seems that they aren't.
Check out the whole list here.