Let $p$ be a prime number and let $G$ be a $p$-group. The Hurwitz space $H$ is the moduli space (or moduli stack) classifying finite Galois covers of (smooth, projective) algebraic curves, with Galois group $G$. Usually one fixes also the genera of the curves in the coverings and some ramification invariants, then $H$ is a smooth variety in characteristic 0. However almost nothing is known about its reduction in characteristic $p$. In this talk I will study the degeneration of the group in a covering, and suggest a definition of a model of $H$ leading to understanding the reduction of $H$.