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A band patch is a patch with a famous band's name, pitcure or sometimes both on it and usually refers to heavy metal bands (Megadeth, Iron Maiden, etc.).
The patches are usually made out of cotton and people who buy them normally sew them on to their demin jackets.
Band patches come in all different shapes and sizes but they are usually square or rectangular.
The most commonly-seen patches are small logo patches and large artwork patches.
The latter are usually reserved for the back of a jacket or coat as they are quite large.