  The cvariant path integral quantization of the theory of the scalar and
spinor particles interacting through the abelian and non-Abelian Chern-Simons
gauge fields is carried out and is shown to be mathematically ill defined due
to the absence of the transverse components of these gauge fields. This is
remedied by the introduction of the Maxwell or the Maxwell-type (in the
non-Abelian case)term which makes the theory superrenormalizable and guarantees
its gauge-invariant regularization and renormalization. The generating
functionals are constructed and shown to be formally the same as those of QED
(or QCD) in 2+1 dimensions with the substitution of the Chern-Simons propagator
for the photon (gluon) propagator. By constructing the propagator in the
general case, the existence of two limits; pure Chern-Simons and QED (QCD)
after renormalization is demonstrated.
  By carrying out carefully the path integral quantization of the non-Abelian
Chern-Simons theories using the De Witt-Fadeev-Popov and the Batalin-Fradkin-
Vilkovisky methods it is demonstrated that there is no need to quantize the
dimensionless charge of the theory. The main reason is that the action in the
exponent of the path integral is BRST-invariant which acquires a zero winding
number and guarantees the BRST renormalizability of the model.
  The S-matrix operator is constructed, and starting from this S-matrix
operator novel topological unitarity identities are derived that demand the
vanishing of the gauge-invariant sum of the imaginary parts of the Feynman
diagrams with a given number of intermediate on-shell topological photon lines
in each order of perturbation theory. These identities are illustrated by an
explicit example.
