The aim is to give an introduction to inverse Galois theory and to some number theoretical topics involved in inverse Galois theory. We will discuss the following topics: the Inverse Galois problems, the geometric approach, the Riemann existence theorem, Hilbert's irreducibility theorem, the Beckmann-Black problem, the Grunwald problem, the Malle conjecture, generic and parametric extensions, etc. NB: this is a basic introduction intended for graduate students or interested colleagues.