  Topological quantum field theories can be used as a powerful tool to probe
geometry and topology in low dimensions. Chern-Simons theories, which are
examples of such field theories, provide a field theoretic framework for the
study of knots and links in three dimensions. These are rare examples of
quantum field theories which can be exactly (non-perturbatively) and explicitly
solved. Abelian Chern-Simons theory provides a field theoretic interpretation
of the linking and self-linking numbers of a link. In non-Abelian theories,
vacuum expectation values of Wilson link operators yield a class of polynomial
link invariants; the simplest of them is the famous Jones polynomial. Other
invariants obtained are more powerful than that of Jones. Powerful methods for
completely analytical and non-perturbative computation of these knot and link
invariants have been developed. In the process answers to some of the open
problems in knot theory are obtained. From these invariants for unoriented and
framed links in $S^3$, an invariant for any three-manifold can be easily
constructed by exploiting the Lickorish-Wallace surgery presentation of
three-manifolds. This invariant up to a normalization is the partition function
of the Chern-Simons field theory. Even perturbative analysis of the
Chern-Simons theories are rich in their mathematical structure; these provide a
field theoretic interpretation of Vassiliev knot invariants. Not only in
mathematics, Chern-Simons theories find important applications in three and
four dimensional quantum gravity also.
