A machine for producing concave or convex aspherical surfaces, comprises a blank carrier spindle mounted for rotation about a first axis, a cutting tool mounted for travelling movement in the direction of the second axis and pivotable about a third axis perpendicular to the first two axes, a control cam, and drive means controlled by the said cam and adapted to displace the said cutting tool, in response to the cam, along its travelling axis between two positions, one of which is an advance position and the other a withdrawn position and either of which is a cutting position and the other a position of rest, the axis of rotation of the spindle and the axis of movement of the tool passing through the centre of the aspherical surface to be cut, and wherein the blank carrier spindle is adjustable in a stable axial position in a spindle support, the control cam is fixed, and the cutting tool is carried by a shoe displaceable on a rotating turret, a drive means being disposed between the said shoe and the said fixed control cam.