Dr. Bertrand Halperin Harvard University Theory predicts the existence of some peculiar phases of quantum condensed matter systems that have multiple degrees of freedom with very low energy, when localized “defects” are introduced. I shall focus on a class of these phases where each defect has half of a conventional degree of freedom, and the defects may be considered as sites for localized zero-energy states of a “Majorana fermion”. Such defects would also exhibit the intriguing property of “non-Abelian statistics” -- i.e., if various defects can be moved around each other, or if two identical defects can be interchanged, the result is a unitary transformation on the quantum mechanical state that depends on the order in which operations are performed but is insensitive to many other details. In my talk, I will try to explain these various concepts and discuss the attempts to realize them in condensed matter systems.