We have developed a process to fabricate nanostructures in oxide materials, especially YBa2Cu3O7-x (YBCO) thin films. The fabrication process is based on a multilayer mask with an amorphous carbon layer as a key component. Electron-beam lithography, reactive ion etching, and ion-beam etching have been the supporting techniques for the process. Nanobridges ranging in width from 50 to 170 nm have been fabricated and characterized. The initial YBCO film was 50 nm thick and had a T-c of 86 K. T-c of the fabricated bridges decreased about 3-5 K with respect to the initial film. The level of the critical current density was 4-9 X 10(5) A/cm(2) at 77 K and 1.6-2 X 10(7) A/cm(2) at 4.2 K. No increase of the critical current density was observed as the cross-section area became smaller. The process has also been applied to fabricate 30-50 nm wide trenches in alpha-axis-oriented YBCO on top of a similar-oriented PrBa2Cu3O7-x (PBCO) layer. No supercurrent was observed in such junctions. The current transport was governed by Mott variable range hopping.