Nevanlinna's value distribution theory, its applications and conjectures

Some new and old problems or conjectures arisen from the studies of Nevanlinna's value distribution theory and its applications to factorization, complex dynamics, and functional equations of meromorphic functions are surveyed, for further investigations. Most of the conjectures were posed by the speaker and his co-workers two or three decades ago, and the latest one is the following conjecture:
Let $f$ be a transcendental entire function and $k$ be a given integer. If $ff^{(k)}$ is periodic, then $f$ itself must be periodic.