This invention provides membrane-mediated electropolishing (MMEP) processes for polishing and/or planarizing metal work pieces using topographically patterned membranes. The processes can be used for both pure metals and alloys, and provide advantages over conventional electropolishing processes and known MMEP processes using smooth membranes. This invention also provides a cathode half-cell and an apparatus useful in membrane-mediated electropolishing processes. The invention also provides processes for electroengraving and electromachining topographic patterns, holes and/or grooves into the surface of a metal work piece.