  The overlap approach to chiral gauge theories on arbitrary $D$--dimensional
lattices is studied. The doubling problem and its relation to chiral anomalies
for $D=2$ and 4 is examined. In each case it is shown that the doublers can be
eliminated and the well known perturbative results for chiral anomalies can be
recovered. We also consider the multi--flavour case and give the general
criteria for the construction of anomaly free chiral gauge theories on
arbitrary lattices. We calculate the second order terms in a continuum
approximation to the overlap formula in $D$ dimensions and show that they
coincide with the bilinear part of the effective action of $D$--dimensional
Weyl fermions coupled to a background gauge field. Finally, using the same
formalism we reproduce the correct Lorentz, diffeomorphism and gauge anomalies
in the coupling of a Weyl fermion to 2--dimensional gravitational and Maxwell
fields.
