  I analyze the interplay of gauge and global symmetries in the theory of
topological defects. In a two-dimensional model in which both gauge symmetries
and {\it exact} global symmetries are spontaneously broken, stable vortices may
fail to exist even though magnetic flux is topologically conserved. Following
Vachaspati and Ach\'ucarro, I formulate the condition that must be satisfied by
the pattern of symmetry breakdown for finite-energy configurations to exist in
which the conserved magnetic flux is spread out instead of confined to a
localized vortex. If this condition is met, vortices are always unstable at
sufficiently weak gauge coupling. I also describe the properties of defects in
models with an ``accidental'' symmetry that is partially broken by gauge boson
exchange. In some cases, the spontaneously broken accidental symmetry is not
restored inside the core of the defect. Then the structure of the defect can be
analyzed using an effective field theory; the details of the physics
responsible for the spontaneous symmetry breakdown need not be considered.
Examples include ``semilocal'' domain walls and vortices that are classically
unstable, but are stabilized by loop corrections, and ``semilocal'' magnetic
monopoles that have an unusual core structure. Finally, I examine the general
theory of the ``electroweak strings'' that were recently discussed by
Vachaspati. These arise only in models with gauge boson ``mixing,'' and can
always end on magnetic monopoles. Cosmological implications are briefly
discussed.
