Supersymmetry (colloquially known as SUSY) is a principle that proposes that each Boson atomic particle (e.g. photons) has a corresponding Fermion particle (e.g. electrons), and vice versa. If such particles were found to exist, several mathematical and logical problems with the current Standard Model of particle physics would be resolved.
“A supersymmetrical extension to the Standard Model would resolve major hierarchy problems within gauge theory, by guaranteeing that quadratic divergences of all orders will cancel out in perturbation theory.”
Despite more than a decade of operation at CERN's Large Hadron Collider (and others) no such particle has yet been found. Supporters of the theory, however, suggest that the new particles may have such large masses that current-day accelerators don't have sufficient energy to create them.