Despite their name, "washerless" faucets do have some washers in them. There are several common types: ball faucets (like those made by Delta), cartridge faucets (like those made by Moen), and ceramic-disc faucets (like those made by American Standard and Kohler). Most have a single handle that controls both hot and cold water. Replacement parts are generally inexpensive and easily available; however, the specific items you'll need and the way they are replaced will depend on the manufacturer. It's usually easiest to buy a replacement kit containing all the washers, springs, O-rings, and other parts you'll need for the make and model of your faucet. Many kits come with instructions to guide you through the process.
To correct water leaking from beneath the handle of a ball faucet, tighten the adjusting ring, a threaded washer that you see when you remove the handle. Shut off the water supply; then close or cover the drain, loosen the screw in the base of the handle, and lift the handle off the spout. Place a putty knife in the slots on top of the washer and turn it clockwise. Replace the handle and test the faucet; if it still leaks, tighten the ring a bit more. Sometimes tightening the ring will make the handle too hard to move. If this happens, unscrew the ring and then the domed metal cap beneath it, and lift out the plastic and rubber cam assembly below the cap. These cam parts can be replaced.
If water leaks from the spout, you'll need to access parts that sit below the cam assembly. The ball, O-rings, seals, and springs are subject to wear and must be replaced periodically. Remove the old ball, and then twist the spout upward to remove it from the faucet's central tube. Once you have exposed the faucet body, pry off the O-rings surrounding it with a screwdriver; then, using needle-nose pliers, remove the small rubber seals and tiny springs that fit into the faucet's inlet holes. The replacement parts must match the old ones exactly, although you can choose a plastic replacement ball or a more expensive brass ball that lasts longer.
Clean all the surfaces and scrape off any mineral deposits before reassembling the faucet. Lubricate the O-rings with plumber's grease before rolling them into their grooves on the faucet body. When you reinstall the spout, push down hard while twisting. Then install the springs and seals, and fit the slot of the ball over the pin in the side of the faucet body, so the ball's ports are aligned with the hot- and cold-water inlets in the faucet when you want the water to flow. Finish by reassembling the cam assembly and handle. If water still leaks from beneath the spout, tighten the cap; if there's a leak from beneath the handle, tighten the adjusting ring as described above.
On a cartridge faucet, pry off the decorative cap and remove the screw underneath; then, lift off the handle and the handle body, a cylindrical slotted fitting. Before removing the spout, unscrew the round retaining nut on top of it.
For leaks from the base of the faucet, remove and replace the O-rings around the faucet body. For other types of leaks, pry out the horizontal metal clip on top of the faucet body, and then pull out the sleeve cartridge from inside it. Buy replacements for the cartridge and the O-rings around it. (Cartridges with brass casings are more expensive, but will last longer.) To install the new cartridge, pull out the stem as far as possible, and press the cartridge into the faucet body. (Water will probably pool around the stem when you do this.) Rotate the cartridge until the protrusions at the top are aligned with the slots in the faucet body for the retainer clip; press down on the cartridge and slide the clip under the protrusions and into the slots.
Position the stem so that its horizontal notch points toward the sink. Then, reassemble the spout and retaining nut. To replace the handle, the ringed end must slide into the handle body and the rear end must hook over the rear of the retaining nut. After you have reinstalled the screw, test the faucet. If the hot and cold water positions are reversed, remove the handle, rotate the stem 180°, and reassemble.
In a ceramic-disc faucet, the cylindrical valve typically contains two discs; when the handle is moved to the "on" position, one disc rotates over another so that holes in both discs align and allow the water to flow. Remove the handle and the locking nut beneath it, and then pry out the cylinder. The valve is made from ceramic, a durable material that seldom needs to be replaced. However, the inlet seals and/or O-rings are subject to wear. Replacement parts can sometimes be difficult to locate, but removing the old ones and inserting the new is reasonably simple.
Once you have replaced all the parts, reassemble the faucet and turn on the water.