The P = NP problem
It is not known whether the classes P and NP are identical.
This is probably the biggest open problem in mathematics.
There are literally hundreds of so-called NP-hard problems.
These are problems in the class NP of which it is know that
if any one of these problems is in the class P, then the
classes P and NP coincide. But nobody has ever found a
polynomial-time algorithm for any of the NP-hard
problems. This is strong empirical evidence for the
conjecture that P is not equal NP, but no proof exists.