  We review the use of the exact renormalization group for realization of
symmetry in renormalizable field theories. The review consists of three parts.
In part I (sects. 2,3,4), we start with the perturbative construction of a
renormalizable field theory as a solution of the exact renormalization group
(ERG) differential equation. We show how to characterize renormalizability by
an appropriate asymptotic behavior of the solution for a large momentum cutoff.
Renormalized parameters are introduced to control the asymptotic behavior. In
part II (sects. 5--9), we introduce two formalisms to incorporate symmetry: one
by imposing the Ward-Takahashi identity, and another by imposing the
generalized Ward-Takahashi identity via sources that generate symmetry
transformations. We apply the two formalisms to concrete models such as QED, YM
theories, and the Wess-Zumino model in four dimensions, and the O(N) non-linear
sigma model in two dimensions. We end this part with calculations of the
abelian axial and chiral anomalies. In part III (sects. 10,11), we overview the
Batalin-Vilkovisky formalism adapted to the Wilson action of a bare theory with
a UV cutoff. We provide a few appendices to give details and extensions that
can be omitted for the understanding of the main text. The last appendix is a
quick summary for the reader's convenience.
