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.