A dicing tape attaching apparatus (10) comprises a fixed table (38) for supporting a mount frame (36), a movable table (31) for supporting a wafer (20) with the reverse surface thereof ground, and a height adjusting unit (70) such as a screw jack for adjusting the height of the movable table. When the dicing tape (3) is attached on the mount frame and the wafer, the height adjusting unit such as the screw jack so operates that the height of the dicing tape attaching surface of the wafer supported on the movable table coincides with the height of the upper surface of the mount frame supported on the fixed table. As a result, the wafer is prevented from forming cuts, cracks and internal distortions.