Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
Number theory is a vast area of mathematics with several different flavours. Core questions that are dealt with in analytic-number-theory are approximating the density of prime-numbers (be it in all or in some residue class modulo a prime) and studying the riemann-zeta-function or more general L-functions. In algebraic number theory, behaviour of ideals under extensions and contractions is one of the central problems. The subject diophantine-equations is the study of integer solutions to certain, mostly polynomial, equations such as Pell's equation. Transcendental number theory is one other challenging subject which starts from transcendence of $e$, $\pi$.
A very popular subject of study, the field of number theory is rich in open problems, such as the riemann-hypothesis, the Ramanujan conjecture, the Goldbach conjecture, and the (non-)existence of odd perfect numbers.