4. We have performed break-junction Josephson-effect measurements in BSCCO and BSCO (Bi₂Sr₂CuO_6+x) single crystals where, for the first time, the Josephson critical current I_c was measured as a function of temperature up to T_c. The large deviations of the I_c(T) from the BCS behaviour have been fully explained by a new model we developed, that takes into account the depression of the order parameter at the surfaces in a high-T_c Superconductor-Insulator-Superconductor (SIS) junction. The SIS depressed-gap model has been extended to the case of SIS' junctions, where S' is a conventional low-Tc superconductor, and the theoretical results have been compared to the experimental I_c(T) obtained in multilevel edge junctions YBa₂Cu₃O_7+x/SrTiO₃/ YBa₂Cu₃O_7+x and in planar tunnel junctions YBa₂Cu₃O_7+x/native barrier/Pb. Both in the planar junctions and in our break junctions, the depressed-gap model can accurately reproduce the deviations of the critical current from the BCS model, both in shape and magnitude, while a model that explains the same I_c(T) deviations in YBCO junctions by assuming an s + id pair symmetry with a dominant d-wave component needs additional hypotheses to do the same.