Step edge grain boundary (GB) junctions and rf-SQUIDs have been made using pulsed laser deposited Y-Ba-Cu-O films on crystalline LaAlO₃ substrates. The steps were developed using various ion-beam etching processes resulting in sharp and ramp type step structures. Sharp step based GB junctions showed behavior of serial junctions with resistively shunted junction (RSJ)-like I-V characteristics. The ramped type step structures resulted in relatively high critical current, I_c, junctions and noisy SQUIDs. The sharp steps resulted in low noise rf-SQUIDs with a noise level below 140 fT/Hz¹2/ down to few Hz at 77 K while measured with conventional tank circuits. The I_c of the junctions and hence the operating temperature range of the SQUIDs made using sharp steps was controlled by both the step height and the junction widths. The junction properties of the SQUIDs were also characterized showing RSJ-like characteristics and magnetic field sensitivities correlated to that of the SQUIDs. Two major low and high background magnetic field sensitivities have been observed for our step edge junctions and the SQUIDs made on sharp steps. High quality step edge junctions with low magnetic field sensitivities made on clean sharp steps resulted in low 1/f noise rf-SQUIDs proper for applications in unshielded environment.