10 unanswered questions (LINK) lay before string theorists. Their solution in terms of String Theory can only help to further understand the fundamental workings of String Theory.
There are obviously more unanswered questions in many different fields. Mathematics has its own set of problems(LINK).
As long as there are fundamental questions to be asked, the foundations will be in jeopardy as a single unexplained flaw or paradox could be a serious flaw in a theory.