  We present a systematic study of the constraints coming from target-space
duality and the associated duality anomaly cancellations on orbifold-like 4-D
strings. A prominent role is played by the modular weights of the massless
fields. We present a general classification of all possible modular weights of
massless fields in Abelian orbifolds. We show that the cancellation of modular
anomalies strongly constrains the massless fermion content of the theory, in
close analogy with the standard ABJ anomalies. We emphasize the validity of
this approach not only for (2,2) orbifolds but for (0,2) models with and
without Wilson lines. As an application one can show that one cannot build a
${\bf Z}_3$ or ${\bf Z}_7$ orbifold whose massless charged sector with respect
to the (level one) gauge group $SU(3)\times SU(2) \times U(1)$ is that of the
minimal supersymmetric standard model, since any such model would necessarily
have duality anomalies. A general study of those constraints for Abelian
orbifolds is presented. Duality anomalies are also related to the computation
of string threshold corrections to gauge coupling constants. We present an
analysis of the possible relevance of those threshold corrections to the
computation of $\sin^2\theta_W$ and $\alpha_3$ for all Abelian orbifolds. Some
particular {\it minimal} scenarios, namely those based on all ${\bf Z}_N$
orbifolds except ${\bf Z}_6$
