  The functional integral computation of the various topological invariants,
which are associated with the Chern-Simons field theory, is considered. The
standard perturbative setting in quantum field theory is rewieved and new
developments in the path-integral approach, based on the Deligne-Beilinson
cohomology, are described in the case of the abelian U(1) Chern-Simons field
theory formulated in S^1 x S^2.
