  The Crane-Yetter state sum is an invariant of closed 4-manifolds, defined in
terms of a triangulation, based on 15-j symbols associated to the category A of
representations over quantum sl2 (at a root of unity). In this thesis, we
define the state sum in terms of a 'PLCW decomposition', which generalizes
triangulations, and generalize A to an arbitrary premodular category. We extend
the state sum to 4-manifolds with corners, making it an extended TQFT. We also
develop a parallel theory based on skeins, which are essentially A-colored
graphs, and we show that the two theories are equivalent. Focusing on the
2-dimensional part, we prove several properties of skein categories, the most
important of which is that they satisfy excision. We provide explicit algebraic
descriptions of the category associated to the once-punctured torus and the
annulus, giving rise to a new tensor product on the Drinfeld center of a
premodular category. As it is well-known that, when A is modular, the
Crane-Yetter state sum computes the signature of a closed 4-manifold, we
connect the Crane-Yetter theory to the signature of a 4-manifold with boundary
and even corners. Finally, we show that the Reshetikhin-Turaev TQFT is a
boundary theory of the Crane-Yetter theory (up to a normalization).
