The influence of antiferromagnetic order on the superconductivity in the non-centrosymmetric heavy fermion compound CePt 3 Si and related materials is discussed. Based on our RPA analysis for the extended Hubbard model two phases could be stabilized by a spin fluctuation induced pairing, with either dominantly p-wave or d-wave symmetry. The antiferromagnetic order plays an essential role for the lowenergy physics, in particular, for the appearance of line nodes in the gap and the enhancement of spin susceptibility below T c . Various properties and possible phase diagrams under pressure are analyzed. The present experimental situation suggests that the p-wave phase is most likely realized in CePt 3 Si.Since the discovery of superconductivity in the noncentrosymmetric heavy Fermion compound CePt 3 Si, 1) superconductivity in materials without inversion center has been attracting growing interest. Many new non-centrosymmetric superconductors with unusual properties have been identified among heavy fermion systems such as UIr, 2) CeRhSi 3 , 3) CeIrSi 3 , 4) CeCoGe 3 , 5) and others like Li 2 Pd x Pt 3Àx B, 6) and KOs 2 O 6 . 7) One immediate consequence of non-centrosymmetricity is the necessity for a revised classification scheme of Cooper pairing states, as parity is not available as a distinguishing symmetry. The pairing states is considered as mixtures of states with different parity imposed by the presence of antisymmetric spin-orbit coupling (ASOC). 8) Recent theoretical studies let to the proposal of various intriguing properties of such a superconductor. [8][9][10][11][12][13][14] In the past the relation between superconductivity and magnetism has been one of the aspects of major interest in heavy fermion systems. Interestingly, all presently known non-centrosymmetric heavy Fermion superconductors, i.e., CePt 3 Si, UIr, CeRhSi 3 , CeIrSi 3 , and CeCoGe 3 , coexist with the magnetism. Although magnetism affects the electronic state profoundly, most of the theoretical studies except for refs. 14 and 15 neglected this aspect so far. The aim of the present study is to elucidate how the magnetism influences the superconducting (SC) phase and how it may be involved in deciding the pairing symmetry in CePt 3 Si. Among the non-centrosymmetric heavy fermion superconductors, CePt 3 Si has been investigated in most detail because the superconductivity occurs at ambient pressure. 1) Although we focus here on CePt 3 Si, we believe that some of our results are qualitatively valid for the other compounds too.In CePt 3 Si superconductivity with T c ¼ 0:75 K appears in the antiferromagnetic (AFM) state with Néel temperature T N ¼ 2:2 K. 1) Neutron scattering measurements characterize the AFM order with an ordering wave vector Q ¼ ð0; 0; Þ and magnetic moments in the ab-plane of the tetragonal crystal lattice. 16) The nature of the SC phase has been characterized by several experiments. The low-temperature properties of the thermal conductivity, 17) superfluid density 18) and specific heat 19) indicate line nodes in the gap,...