A cutting apparatus for cutting a longitudinally extending member such as rail comprises a frame; means for clamping the frame to the member to be cut; and a cutting torch which is pivotally attached to the frame. A pivoted torch holder is provided having bearing means permitting axial displacement of the torch relative to the member to be cut. The precise axial distance of the tip of the torch relative to the member to be cut is determined throughout the making of the cut by means comprising a plate attached to the frame and defining a cam surface and a cam follower connected to the torch. The shape of the cam surface thus makes it possible for the path described by the tip of the torch to conform, if desired, to the profile of the member to be cut.