  Theoretical issues of exact chiral symmetry on the lattice are discussed and
related recent works are reviewed. For chiral theories, the construction with
exact gauge invariance is reconsidered from the point of view of domain wall
fermion. The issue in the construction of electroweak theory is also discussed.
For vector-like theories, we discuss unitarity (positivity), Hamiltonian
approach, and several generalizations of the Ginsparg-Wilson relation
(algebraic and odd-dimensional).
