  We develop a microscopic and gauge-invariant theory for collective modes
resulting from the phase of the superconducting order parameter in
non-centrosymmetric superconductors. Considering various crystal symmetries we
derive the corresponding gauge mode $\omega_{\rm G}({\bf q})$ and find, in
particular, new Leggett modes $\omega_{\rm L}({\bf q})$ with characteristic
properties that are unique to non-centrosymmetric superconductors. We calculate
their mass and dispersion that reflect the underlying spin-orbit coupling and
thus the balance between triplet and singlet superconductivity occurring
simultaneously. Finally, we demonstrate the role of the Anderson-Higgs
mechanism: while the long-range Coulomb interaction shifts $\omega_{\rm G}({\bf
q})$ to the condensate plasma mode $\omega_{\rm P}({\bf q})$, it leaves the
mass $\Lambda_0$ of the new Leggett mode unaffected and only slightly modifies
its dispersion.
