These theories solve some deep theoretical problems about the grand unified theories, but predict that EVERY known particle has a partner supersymmetric particle that is more massive.
No supersymmetric particle has yet been seen, but experiments underway at CERN beginning in Autumn 1995 will search for the partner of the W boson, and experiments at Fermilab are looking for the partners of the quarks and gluons.
One of the supersymmetric particles (the partner of the photon) might be the missing dark matter.