  We only require generalized chiral symmetry and $\gamma_5$-hermiticity, which
leads to a large class of Dirac operators describing massless fermions on the
lattice, and use this framework to give an overview of developments in this
field. Spectral representations turn out to be a powerful tool for obtaining
detailed properties of the operators and a general construction of them. A
basic unitary operator is seen to play a central r\^ole in this context. We
discuss a number of special cases of the operators and elaborate on various
aspects of index relations. We also show that our weaker conditions lead still
properly to Weyl fermions and to chiral gauge theories.
