Abstract: Fix $g \in \mathbb{N}$ such that $g$ is not a perfect square. Artin's primitive root conjecture (1927) states that there exist infinitely many primes $p \in \mathbb{N}$ such that $g$ generates the finite cyclic group $(\mathbb{Z}/p\mathbb{Z})^{\times}$. Nearly a century later, Artin's conjecture remains wide-open: in fact there is no known specified $g$ for which the conjecture has been resolved.

In this talk, we survey the interesting history of Artin's conjecture and introduce several new variants to the problem. Specifically, we discuss an "Artin Twin Primes Conjecture"; and prove an appropriate analogue of Artin's conjecture for algebraic function fields.