Place dressed fish belly toward you. With a very sharp knife, cut through flesh from end of cavity back through to tail.
Place knife blade against backbone and cut along backbone from head end to tail on one side of fish, severing ribs and top piece from backbone.
Lay to piece aside. Remove backbone from remaining side.
With a smaller knife, trim away rib and fin bones from both pieces. Pull out pin bones, if desired.
If you wish to skin fillets, place skin-side-down on cutting surface. Hold tail end tightly. With sharp knife, cut down through the flesh to skin. Flatten knife against skin and cut flesh away by sliding it toward head end while holding tail end of skin firmly.
Prepared salmon fillets can be baked, poached or grilled, or cut into serving sized portions.