Non-BPS non-Abelian vortices with CP^1 internal moduli space are studied in
some theories with gauge group SU(2) X U(1) in the Higgs phase, both in a
supersymmetric and in a non-supersymmetric setting.
The distinctive feature of a non-Abelian vortex is the presence of
an internal CP^1 space of orientational degrees of freedom.
For fine-tuned values of the couplings the vortices are BPS
and there is no net force between two static parallel vortices
at arbitrary distance.
For generic values of the couplings the interactions between two vortices
depend non-trivially on their relative internal orientations.
In additition to the Type I/Type II regimes typical of Abelian
superconductors, some new regimes are found.
Unlike the type I (type II) case, where the interaction is always
attractive (repulsive), the type I^* and II^* have both attractive
and repulsive interactions depending on the relative CP^1 orientation. In
the supersymmetric setting that we considered
we have found just the Type I and the Type I^* regimes.
A rich variety of interactions is found at small vortex separation.
For some values of the couplings we found evidence of the existence of a
bound state of two static vortices at a non-zero distance.
