We propose a formulation of lattice fermions with one-sided difference
s that is hermitian, chirally symmetric (barring a bare mass term) and
completely free of doubling. To obtain the axial anomaly in perturbat
ion theory it was necessary to break chiral symmetry on the lattice on
ly through a bare mass term for the physical fermion. The chiral limit
may be taken once the continuum limit is reached. We comment on the r
ole of the mass term with examples elsewhere in field theory.