A duality symmetry between electric and magnetic fields in the vacuum Maxwell equations suggest that electric monopoles should have a magnetic counterpart. Elusiveness of magnetic monopoles has been a serious challenge to physics for a century both at the theoretical and experimental side. The search for monopoles as energy frontiers are crossed always gets a boost in various detectors e.g. the MoEDAL experiment at the LHC, IceCube and ANTARES. At colliders, monopoles could be produced in pairs and would be highly ionizing particles.
Since we have not found them we should be open minded about what they could be and what they may be made of. I shall give the classic construction of a fundamental magnetic monopole due to Dirac which was based on an Abelian (U(1)) gauge theory but had infinite energy. The Dirac monopole was not just a monopole but had a string stuck to it (like a tadpole) but a string that we cannot physically detect but which leads to charge quantization. I shall then give the construction of a magnetic monopole due to 't Hooft and Polyakov who based the theory on a non-Abelian gauge symmetry. The latter proposal produced a finite energy soliton as the monopole. Proposals for such monopoles typically occur in Grand Unified Theories (GUT) and have masses of the order of the GUT scale and so their possible observation will be in the astrophysical domain. Within the context of the electroweak Standard Model (SM), we shall describe a possible monopole at the TeV scale and discuss how its divergent energy can be renormalized by effects derived from physics beyond the SM (BSM).
A recent challenge to the hegemony of gauge theories in the study of magnetic monopoles has been inspired by BSM physics. I shall review the construction of such monopoles which involves new elements such as torsion, a dilaton and non-electroweak Higgs fields. A range of parameters will be shown to allow TeV scale masses for these monopoles.