A profiling apparatus comprising a profiling member (33A, 33B) for profiling a workpiece and a nozzle body for outputting water or a laser beam, and cutting the workpiece by water or a laser beam from the nozzle body while the workpiece is profiled by the profiling member, comprising: a multi-joint robot that has an arm; a floating mechanism that has a base portion fixed to a tip of the arm, a slider supported through a linear bearing by the base portion so as to be slidable in predetermined first and second directions (X1, X2), a stay fixed to the slider and supporting the nozzle body at two points of the front edge of the stay which are spaced from each other by an interval in vertical direction; and characterised by: two air cylinders jointed to the slider, pressing the slider in the first direction, and push back the slider so as to set the slider to a floating state when air pressure applied to the second direction is equal to or more than a predetermined pressure; and displacement means that has a servo motor fixed to a support plate fixed to the stay, and a slider jointed to an output shaft of the servo motor by a ball screw, and configured to make the profiling member (33A, 33B) to profile a profile target surface of a workpiece by the function of the servo motor to change positional relationship between the profiling member and the nozzle body. a multi-joint robot that has an arm; a floating mechanism that has a base portion fixed to a tip of the arm, a slider supported through a linear bearing by the base portion so as to be slidable in predetermined first and second directions (X1, X2), a stay fixed to the slider and supporting the nozzle body at two points of the front edge of the stay which are spaced from each other by an interval in vertical direction; and characterised by: two air cylinders jointed to the slider, pressing the slider in the first direction, and push back the slider so as to set the slider to a floating state when air pressure applied to the second direction is equal to or more than a predetermined pressure; and displacement means that has a servo motor fixed to a support plate fixed to the stay, and a slider jointed to an output shaft of the servo motor by a ball screw, and a base portion fixed to a tip of the arm, a slider supported through a linear bearing by the base portion so as to be slidable in predetermined first and second directions (X1, X2), a stay fixed to the slider and supporting the nozzle body at two points of the front edge of the stay which are spaced from each other by an interval in vertical direction; and characterised by: two air cylinders jointed to the slider, pressing the slider in the first direction, and push back the slider so as to set the slider to a floating state when air pressure applied to the second direction is equal to or more than a predetermined pressure; and displacement means that has two air cylinders jointed to the slider, pressing the slider in the first direction, and push back the slider so as to set the slider to a floating state when air pressure applied to the second direction is equal to or more than a predetermined pressure; and displacement means that has a servo motor fixed to a support plate fixed to the stay, and a slider jointed to an output shaft of the servo motor by a ball screw, and configured to make the profiling member (33A, 33B) to profile a profile target surface of a workpiece by the function of the servo motor to change positional relationship between the profiling member and the nozzle body.