The invention relates to a transport device (38) for transporting a workpiece (23) between a clamping chuck (22) and a workpiece carrier (30). The transport device (38) comprises a gripper arrangement (39) with a pivot arm (40). One end of the pivot arm (40) is supported on the machine tool (20) and, in particular, a machine slide (25) via a first pivot drive (41) so as to be pivotable about a first pivot axis (S1). On the other end, the pivot arm (40) supports a second pivot drive (42) that defines a second pivot axis (S2). A workpiece gripper (43) is supported so as to be pivotable about this second pivot axis (S2). The two pivot axes are inclined by 45° relative to each other. A workpiece (23) held by the workpiece gripper (43) has a workpiece longitudinal axis (W) that extends in a first position (I) of the workpiece gripper (43) at a right angle with respect to the pivot axis (S1). In a second position (II) of the workpiece gripper (43), the workpiece longitudinal axis (W) is oriented parallel to the first pivot axis (S1). The workpiece longitudinal axis (W) of the workpiece (23) held by the workpiece gripper (43) is inclined by 45° relative to the second pivot axis (S2).