Abstract. Let K be a finite Galois extension of the rationals and rho a complex representation of the Galois group Gal(K/Q). In 1923, Artin attached to this data an L-function, L(s,rho), which he conjectured has analytic continuation to the complex plane and satisfies a functional equation relating s to 1-s. Through the development of class field theory and a theorem of Brauer (1947), we know today that Artin's L-functions are meromorphic in the plane. However, despite this and more recent progress related to the Langlands program, the full conjecture remains largely unsolved. I will survey what is known about Artin's conjecture, why it is important, and address the simpler question of whether specific instances of the conjecture can be verified in finite time.