We study various modifications to the minimal models of gauge-mediated
supersymmetry breaking. We argue that, under reasonable assumptions,
the structure of the messenger sector is rather restricted. We investi
gate the effects of possible mixing between messenger and ordinary squ
ark and slepton fields and, in particular, violation df universality.
We show that acceptable values for the mu and B parameters can natural
ly arise from discrete, possibly horizontal, symmetries. We claim that
in models where the supersymmetry-breaking parameters A and B vanish
at the tree level, tan beta could be large without fine-tuning. We exp
lain how the supersymmetric CP problem is solved in such models.