In an effort to determine the suitability of Josephson junctions for applications in quantum computation, we measured low-noise dc current-voltage characteristics of Nb/AlO/sub x//Nb and Al/AlO/sub x//Al junctions in the temperature range from 90 mK to 1 K. Nb-based samples were obtained from several different facilities with critical currents of a few /spl mu/A and critical current densities between 100 and 3000 A/cm/sup 2/. We fabricated the Al-based samples using double-angle evaporation and obtained critical currents of a few /spl mu/A with critical current densities of about 30 A/cm/sup 2/. We found that the sub-gap leakage current in the Nb-based samples does not depend on temperature in the range 90 mK to 1 K, whereas that for the Al-based samples follows the expected BCS behavior to about 150 mK. Our Al-based samples have a lower level of dissipation than Nb-based devices; however, both Al- and Nb-based samples achieved dissipation levels sufficiently low for some quantum computing applications.