The magnetic monopole
problem is that it really offends
our artistic sensibilities
that they don't seem to exist
. Classically, Maxwell's Equations
are completely symetric for electric fields and magnetic fields, except that the source term
for the magnetic field
is always zero.
In special relativity, we find that magnetic fields are just Lorenz Transformations of electric fields, and thus there is no compelling need for monopoles. If the speed of light were infinite, there wouldn't even be any magnetic fields.
String Theory and other grand unified theories again have a beautiful mathematical symmetry showing the possibility of existence of magnetic monopoles. Also, theories of dark matter and other cosmological theories involve magnetic monopoles.
In short, magnetic monpoles are beautiful and elegant, but don't seem to exist. Mother nature is a bitch.