Superconductive LSIs with Josephson junctions have features such as low power dissipation and high switching speed. In this paper, we review our developed 4-Kbit RAM with vortex transitional memory cells as an illustration of superconductive LSIs with Josephson junctions. We have developed a fabrication process technology for the 4-Kbit RAM. In the 4-Kbit RAM, 380ps access time and 9.8 mW power dissipation have been experimentally obtained. And also, we have estimated a suitable moat structure to reduce the influence of trapped magnetic structure. The 4-Kbit RAM has been successfully operated with a bit yield of 99.8%. Furthermore, we discuss GHz testing which is one of the most significant issues concerning superconductive digital LSIs.