  We give a prescription for embedding classical solutions and, in particular,
topological defects in field theories which are invariant under symmetry groups
that are not necessarily simple. After providing examples of embedded defects
in field theories based on simple groups, we consider the electroweak model and
show that it contains the $Z$ string and a one parameter family of strings
called the $W(\alpha )$ string. It is argued that, although the members of this
family are gauge equivalent when considered in isolation, each member should be
considered distinct when multi-string solutions are considered. We then turn to
the issue of stability of embedded defects and demonstrate the instability of a
large class of such solutions in the absence of bound states or condensates.
The $Z$ string is shown to be unstable when the Weinberg angle ($\theta_w$) is
$\pi /4$ for all values of the Higgs mass. The $W$ strings are also shown to be
unstable for a large range of parameters. Embedded monopoles suffer from the
Brandt-Neri-Coleman instability. A simple physical understanding of this
instability is provided in terms of the phenomenon of W-condensation. Finally,
we connect the electroweak string solutions to the sphaleron: ``twisted'' loops
of W string and finite segments of W and Z strings collapse into the sphaleron
configuration, at least, for small values of $\theta_w$.
