  Superconductivity and magnetism in the non-centrosymmetric heavy fermion
compound CePt$_3$Si and related materials are theoretically investigated. Based
on the randam phase approximation (RPA) analysis for the extended Hubbard model
we describe the helical spin fluctuation induced by the Rashba-type
anti-symmetric spin-orbit coupling and identify the two stable superconducting
phases with either dominantly p-wave ($s$ + $P$-wave) or d-wave ($p$ + $D$ +
$f$-wave) symmetry. The influcnce of the coexistent anti-ferromagnetic order is
investigated in both states. The SC order parameter, quasiparticle density of
state, NMR $1/T_{1}T$, specific heat, anisotropy of $H_{\rm c2}$ and possible
multiple phase transitions are discussed in details. The comparison with
experimental results indicates that the $s$ + $P$-wave superconducting state is
likely realized in CePt$_3$Si.
