Naturalness bounds on gauge mediated soft terms

After a general discussion about the quantitative meaning of the naturalness upper bounds on the masses of supersymmetric particles, we compute these bounds in models with gauge-mediated soft terms. We find interesting upper limits on the right-handed slepton masses that, unless the messenger fields are very light, disfavor minimal models with large messenger content. Deep unphysical minima, that however turn out to be not dangerous, are usually present in such models. The mu-problem can be solved by adding a light singlet only at the price of a large amount of fine tuning that gives also rise to heavy sparticles and large tan\beta.