  Modular tensor categories are generalizations of the representation
categories of quantum groups at roots of unity axiomatizing the properties
necessary to produce 3-dimensional TQFTs. Although other constructions have
since been found, quantum groups remain the most prolific source. Recently
proposed applications to quantum computing have provided an impetus to
understand and describe these examples as explicitly as possible, especially
those that are "physically feasible." We survey the current status of the
problem of producing unitary modular tensor categories from quantum groups,
emphasizing explicit computations.
