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The fez is a red short cylinder with a tassel on the top of it. It was popularized during Ottoman period, but still is an icon of Ottoman culture. Fez was used in the Middle East, Eastern Europe, and North Africa, so it's unclear where were its origins. It was a regular part of uniforms of many countries and today is mostly exotic fashion item and part of national costumes.
Sombrero is a Mexican hat famous for its very wide brim, high and pointed crowns and a slightly upturned edge. Over the last several centuries, sombrero and poncho managed to become the iconic part of the Mexican society. Today this hat is worn not only in Mexico but also all over the world.
One of the most famous types of hats is fedora. Combination of practicality and style, it lasted long and is still worn, not in such great numbers as when it appeared but it is still here. Read more about fedora here.
Famous bowler hat, symbol of working but also upper classes as well, a better cowboy hat than a cowboy hat started as a hard hat for gamekeepers. You can read more about history of bowler hats here.