The results that I will present are part of a program, pursued by my students N. Ghazi, F. Legrand and myself, on the Galois extensions E/Q that can be obtained from a Galois extension F/Q(T) by specializing T. Beyond the Galois group of the specialized extensions, we have investigated their local behavior (inertia groups, decomposition groups) at given primes. I will also explain the implications of the results on classical topics from Inverse Galois Theory, like the Regular Inverse Galois Problem, the Noether program, the Beckmann-Black problem, the Grunwald Problem, the Inverse Galois Problem, etc.