The work I will talk about is motivated by the Regular Inverse Galois Problem: show that every finite group G is the Galois group of a Galois extension F/Q(T) with Q algebraically closed in F. I will discuss two types of results. First, some strong variants of the RIGP related to the notion of parametric extensions, which will be shown to fail. Second, a strong consequence of the RIGP related to a conjecture of Malle on the number of Galois extensions with a given group and with bounded discriminant.