The Malle conjecture predicts that the number of Galois extensions of Q with given group G and discriminant bounded by some real number y > 0 grows like y^a, for some exponent a > 0. This statement is known for nilpotent groups. The work I will present establishes it for Sn, An, many simple groups and more generally all regular Galois groups overQ. The constructed extensions can be further requested to satisfy some notable local conditions. Our method uses a new version of Hilbert's Irreducibility Theorem that counts specialized extensions and not just the specialization points. A new ingredient is the self-twisted cover that we will introduce.