  We present a general formulation of chiral gauge theories, which admits Dirac
operators with more general spectra, reveals considerably more possibilities
for the structure of the chiral projections, and nevertheless allows
appropriate realizations. In our analyses we use two forms of the correlation
functions which both also apply in the presence of zero modes and for any value
of the index. To account properly for the conditions on the bases the concept
of equivalence classes of pairs of them is introduced. The behaviors under
gauge transformations and under CP transformations are unambiguously derived.
