An apparatus for measuring a surface of a workpiece has a multi-link articulated arm and a roughness sensor carried by the arm, which includes a sensing element linearly displaceable along an advance direction and elastically deflectable along a deflection direction, and which has a coupling link to connect it to a movable carrier of a coordinate measuring apparatus or of a robot. A first arm portion is rotatable relative to the coupling link about a first axis of rotation. A second arm portion is rotatable relative to the first arm portion about a second axis of rotation and arranged between the first and third arm portion which is rotatable relative to the second arm portion about a third axis of rotation, and to which the roughness sensor is fastened. The deflection direction is arranged parallel to the third axis while the sensing element is displaced linearly along the advance direction.