Superconductivity in materials without inversion symmetry in the respective crystal structures occurs in the presence of antisymmetric spin-orbit coupling as a consequence of an emerging electric field gradient. The superconducting condensate is then a superposition of spin-singlet and spin-triplet Cooper pairs. This scenario accounts for various experimental findings such as nodes in the superconducting gap or extremely large upper critical magnetic fields. Spin-triplet pairing can occur in non-centrosymmetric superconductors in spite of Anderson's theorem that spin-triplet pairing...