  Non-Abelian vortices in six spacetime dimensions are obtained for a
supersymmetric U(N) gauge theory with N hypermultiplets in the fundamental
representation. Massless (moduli) fields are identified and classified into
Nambu-Goldstone and quasi-Nambu-Goldstone fields. Effective gauge theories for
the moduli fields are constructed on the four-dimensional world volume of
vortices. A systematic method to obtain the most general form of the effective
Lagrangian consistent with symmetry is proposed. The moduli space for the
multi-vortices is found to be a vector bundle over the complex Grassmann
manifold.
