The Hurwitz moduli space theory provides an arithmetic-geometric approach to the regular inverse Galois problem: the question is reduced to finding rational points on varieties. Patching techniques can be used over p-adic fields, and extend to the realization of profinite groups. In this context, the moduli aspect leads to towers of varieties, for which there are new specific arithmetic issues. There are positive results over p-adic fields, negative results over number fields, and over other fields like large fields, the situation is unclear.