Prime numbers, Diophantine equations, analytic number theory, 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. 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 $e$, $\pi$.
A very popular subject of study, the field of number theory is rich in open problems, such as the Riemann hypothesis, the Ramanaujan conjecture, the Goldbach conjecture, and the (non)existence of odd perfect numbers.