A finishing machine (100) for finish machining of curved workpiece surfaces on workpieces (200) has a roundtable (110) that can be rotated cyclically about a roundtable axis (112), as well as a plurality of workpiece spindles (120, 130, 140, 150), which are arranged at a specified angular pitch on the roundtable. Each workpiece spindle is rotatable about a spindle axis by means of a spindle drive and has a workpiece holder for holding a workpiece in such a manner that the workpiece held by the workpiece holder can be rotated about a workpiece axis by rotation of the workpiece spindle. The finishing machine additionally has at least one finishing unit comprising a pressing device for pressing an abrasive-studded machining tool against a curved peripheral surface of the workpiece with a pressing force directed radially to the workpiece axis. A workpiece held by the workpiece holder can be moved into a machining position on the finishing unit by rotation of the roundtable and, after completion of the finish machining, can be moved from this machining position to a subsequent workstation. A device for active support of the workpiece against transverse forces is provided on at least one workpiece section with axial separation from the workpiece holder.