The Jones polynomial and the Kauffman bracket are constructed, and the
ir relation with knot and link theory is described. The quantum groups
and tangle functor frameworks for understanding these invariants and
their descendents are given. The quantum group U-q(sl(2)), which gives
rise to the Jones polynomial, is constructed explicitly. The 3-manifo
ld invariants and the axiomatic topological quantum field theories whi
ch arise from these link invariants at certain values of the parameter
are constructed and proven to be invariant.