In the 1920's, Emil Artin defined his non-abelian L-series and conjectured that these L-series extend holomorphically to the entire complex plane (apart from a possible pole at s=1). This is called Artin's holomorphy conjecture. Later, he indicated how these L-series could be used to solve some classical problems of analytic number theory. To illustrate, he formulated his famous primitive root conjecture. Artin's conjectures are still open. However, tremendous progress has been made so far on both of them and this talk will survey some of these developments, more than a century later.