  We present a discussion of embedded vortices in general Yang-Mills theories.
The origin of a family structure of solutions is shown to be group theoretic in
nature and a procedure for its determination is developed. Vortex stability can
be quantified into three types: Abelian topological stability, non-Abelian
topological stability, and dynamical stability; we relate these to the family
structure of vortices, in particular discussing how Abelian topological and
dynamical stability are related. The formalism generally encompasses embedded
domain walls and embedded monopoles also.
