  We address the problem of non-linear transport through discrete electronic
levels in a small quantum dot coupled to superconducting electrodes. In our
approach the low temperature I-V characteristics can be calculated including
all multiple quasi-particle and Andreev processes. The limit of very weak
coupling to the leads and large charging energies is briefly analyzed comparing
the calculated lineshapes of the I-V curves with recent experimental results.
When the coupling to the leads increases and Coulomb blockade effects can be
neglected, the combination of multiple Andreev processes and resonant
transmission gives rise to a rich subgap structure which largely differs from
the one found in the more studied S-N-S systems. We show how multiple processes
can be included within a simple sequential tunneling picture qualitatively
explaining the subgap structure. We suggest an experimental set-up where the
predicted effects could be observed.
