  Supersymmetric theories involving a spontaneously broken flavor symmetry can
lead to fermion masses which vanish at tree level but are generated by
radiative corrections. In the context of supersymmetric theories with minimal
low energy field content we discuss which fermion masses and mixings may be
obtained radiatively, and find that constraints from flavor changing
phenomenology imply that only the first generation fermion masses and some (but
not all) CKM mixings can naturally come from radiative corrections. We also
consider general conditions on theories of flavor which guarantee the existence
of tree level massless fermions while having non-trivial CKM matrix elements at
tree level. Two complete models of flavor are presented. In the first model,
all first generation fermion masses are radiatively generated. In the second
model, the electron and up quark mass are due to radiative corrections whereas
the down mass appears at tree level, as does a successful prediction for the
Cabibbo angle $\sin \theta_c = \sqrt{m_d/m_s}$.
