We fabricated and characterized YBa2Cu3Oy step-edge Josephson junctions and DC SQUID's on MgO (001) substrates. Sharp steps on MgO (001) were obtained by the techniques of photolithography and Ar-ion milling with step heights varying from 300 to 350 nm. The current-voltage (I-V) characteristic of the junction is consistent with the resistively shunted junction (RSJ) model. We obtained Jc=1.0×105 A/cm2 and IcRn=110 μV at 77 K. In addition to the conventional Shapiro steps resulting from the AC Josephson effect, half-integral steps were observed. The width of these half-integral steps was sensitive to the RF power and the magnetic field. DC SQUID's have been operated below 78 K and the observed voltage modulation agrees with the expected value.