My shower has three knobs - hot, cold and diverter. I recently replaced the hardware and the cold water will not turn off. There is a constant flow of water coming out. All washers are fine. What is causing the leakage?

When you say hardware are you talking only the knob and escutcheon or what? Did you replace a cartridge or washer?

The seat is damaged or you left an old washer in there. Are you sure the valve is closed all the way?

I replaced the whole handle unit - stem, washer, handle. The surface where the washer makes contact seems to be fine. Only the cold water will not shut off.

Remove the cold water stem again. Reinstall with the stem in the open position. In other words, hold the body of the stem and turn the stem itself to open position and then reinstall.

When I replaced the handle assembly I loosened the seat. I replaced the seats. No more leaking.
Always loosen the handle stem before extracting the assembly from the pipe.