The questions that are dealt with in number theory (analytic for instance) are approximating the density of prime numbers (be it in all or in some residue class modulo a prime), studying the zeta function which is the central aspect in analytic number theory. In algebraic number theory behaviour of ideals under extensions and contractions is one of the central problems. Diophantine equations study integer solutions to certain polynomial equations such as Pell's equation. Transcendental number theory is one other challenging subject which starts from transcendence of $e$, $\pi$ and has now developed well too!
A very popular subject of study, this is rich in open problems. For instance the Goldbach conjecture, the existence of odd perfect numbers, to name some.