In 1927, Artin formulated his famous conjecture on primitive roots. The most basic question, which is still open, is as follows. For an odd prime number \(p\), we say that \(2\) is a primitive root modulo \(p\) if every integer that is not a multiple of \(p\) is congruent to a power of \(2\) modulo \(p\). Are there infinitely many primes \(p\) such that \(2\) is a primitive root modulo \(p\)? Artin's conjecture (that has been proven by Hooley in 1967 under GRH) says in particular that there is a positive density of primes \(p\) such that \(2\) is a primitive root modulo \(p\). This density is conjectured to be Artin's constant, which is roughly \(37/%\). Beyond the classical results, I will present very general results on the index map (joint work with Järviniemi and Sgobba), an unexpected lower bound on the density (joint work with Shparlinski) and a computational project (joint work with Tholl).-- This seminar is part of the activities of the Dipartimento di Eccellenza CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.