Semiconductor wafers are produced by a process of: a) providing a semiconductor wafer by cutting a silicon ingot into wafers; b) rounding the edge of the wafer, so that the wafer comprises plane surfaces on the frontside and backside and rounded oblique surfaces in the edge region; c) polishing the frontside and backside of the wafer, the frontside being polished by chemical-mechanical polishing using a polishing pad which is free of abrasive fixed in the polishing pad; backside polishing being carried out in three steps, using a polishing pad containing fixed abrasive which is pressed onto the backside of the wafer, a polishing agent free of solids introduced between the polishing pad and the backside of the wafer in the first step, a polishing agent containing abrasive being introduced in the second and third steps, a polishing pressure of 8-15 psi in the first and second steps being reduced to 0.5-5 psi in the third step.