Electrically charged particles, such as the electron, are ubiquitous.  By contrast, no elementary particles with a net magnetic charge have ever been observed, despite intensive and prolonged searches.

An alternative strategy is to realise them not as elementary but rather as emergent particles, i.e., as manifestations of the correlations present in a strongly interacting many-body system.  Indeed, it turns out  that magnetic monopoles do emerge in a class of exotic magnets known collectively as spin ice: the dipole moment of the underlying electronic degrees of freedom fractionalises into monopoles!  These monopoles can in principle be detected in an experiment modelled after the celebrated Stanford search for magnetic monopoles of cosmic origin.  The many-body state of spin ice, in turn, represents a  new type of  phase outside the standard classification of magnetic order and disorder.

Reference: C. Castelnovo, R. Moessner and S. L. Sondhi, Nature 451, 42 (2008).