Finish polishing of optics with magnetic media has evolved extensively over the past decade. Of the approaches conceived during this time, the most recently developed process is called magnetorheological finishing (MRF). In MRF, a magnetic field stiffens a fluid suspension in contact with a workpiece. The workpiece is mounted on the rotating spindle of a computer numerically controlled machine. Driven by an algorithm for machine control that contains information about the MRF process, the machine deterministically polishes out the workpiece by removing microns of subsurface damage, smoothing the surface to a microroughness of 10 angstroms rms, and correcting surface figure errors to less than 0.1 micrometers p-v. Spheres and aspheres can be processed with the same machine set-up using the appropriate machine program. This paper describes MRF and gives examples which illustrate the capabilities of a pre-prototype machine located at the Center for Optics Manufacturing.