Let R be a complete discrete valuation ring of mixed characteristic (0, p) with fraction field K. The stable reduction theorem affirms that given a smooth, projective, geometrically connected curve over K, C/K, with genus \geq 2, there exists a unique finite Galois extension M/K minimal for the inclusion relation such that C_{M}:= C x M has stable reduction over M. A such extension is called monodromy extension of C/K and the Galois group Gal(M/K) is called the monodromy group of C/K. In this paper, the authors study stable models of p-cyclic covers of P^1_K. At first, they work with covers of arbitrarily high genus having potential good reduction. In particular, they determine for such covers the monodromy extension, the monodromy group, its filtration and the Swan conductor. Successively, the authors restrict their attention to the case p = 2 and g = 2.