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If you have a pump-action shotgun, twist off the end-cap located at the top of the pump, right next to the barrel. Sometimes the barrel will stick – if your barrel is sticking, slide the pump arm all the way down and then open the breach, which will release the barrel.  Twist the barrel off away from the gun itself. For single-shot shotguns, take off the forearm and then remove the barrel. For semi-automatic shotguns pull the action all the way back. Unscrew the forearm cap and then slide the forearm off. Remove the thick, caked-on carbon buildup, which is created by the friction of use. Inspect and wipe carefully to remove any traces of old oil and all unburnt powder buildup. Clean all the gaps well.  Be sure to wipe the ejector and the area around the chamber, too. You will find that after cleaning certain areas, your cloth will be black. Clean these areas more until this no longer happens. It is better to spray more solvent than you need rather than not enough. Make sure to cover all areas and focus on those that are visibly caked with dirt and buildup. After generously applying the solvent, allow it to sit for 15 minutes so that the dirt and buildup are well-soaked.  There are many solvent products on the market. Some are quite toxic, so be sure to read instructions and warnings on the bottle before using. If recommended by the manufacturer, use a face mask to reduce inhalation of toxicants and wear gloves. Some solvent products are safe for skin contact. If possible, look for these products and use those for your gun cleaning. M-Pro 7 is one such product. You can buy special brushes made for this, but a soft-bristled toothbrush will also work great. Scrub the surface thoroughly, working in the solvent, which helps loosen the buildup on the gun. Be thorough and try to get into all the nooks and crannies with your brush. In this instance, a completely lint-free cloth allows for the most effective cleaning. Wipe the entire surface of the gun, removing all of the dirt and solvent. Concentrate on any areas that turn your cloth black and wipe until no more black appears. Spray the solvent all over the gun, just as you did before, and wipe it down thoroughly again. Concentrate on any areas that turn your cloth black and strive to get into the nooks and crannies. This second wipe-down will get the surface of your gun pristine. Gun cleaning picks are usually made of high-strength polymer, which will not scratch the surface of your gun. The most common area for carbon deposits is in the chamber, and the most effective way of cleaning this is with a pick.  Gun cleaning picks are tools with different kinds of hooked and pointed ends designed to clean the really hard to reach areas of your gun.  They look a lot like the dental picks your dentist uses during a routine tooth cleaning. Be sure to also clean in the corners and sides of the metal, since there are other areas where buildup often occurs. Use a cleaning rod to swab the cloth into the bore. Repeat this action with more clean, solvent-soaked cloths until the clothes come out with no visible residue on them. Then use a dry cloth to swab any remaining dirt from the barrel.  Soak another clean cloth in oil and swab the barrel with this, which will protect it from rust. Instead of using a cleaning rod and cloth to clean the barrel, try a reusable foam swab made particularly for gun cleaning. This looks like a household Q-Tip except much larger, and you can remove the tip, wash it, and reuse it.  For a less precise cleaning on a pump-action shotgun, run a bore snake through the barrel. This is a good instrument to use when your gun doesn't need a deep cleaning, but rather a quick touch up.

Summary:
Remove the barrel. Wipe down all components with a clean cloth. Spray solvent liberally on all components. Scrub all over the surface of the gun with a soft-bristled brush. Wipe the gun down with a lint-free cloth. Soak the gun insolvent again. Use a gun cleaning pick to clean off any carbon and powder buildup. Swab down into the barrel using a clean, solvent-soaked cloth.