The folding fan was invented in Japan in the 6th century AD, and the idea then spread to Korea and China. By the 15th century, the Chinese were importing vast quantities of Japanese fans and by the 18th century wealthy European ladies carried one, as they were both beautiful and practical, and a whole court etiquette of flirtation was built around fans.
This is a 19th century folding fan by Katsushika Isai which is owned by New York's Metropolitan museum:
The fans are made of wood, thread and painted paper - in traditional Japanese society, the higher you ranked, the more strips of wood in your fan.
Fans have fallen out of fashion in the west thanks to air conditioning, but the Japanese royal family always carry one on official occasions. Here is the Empress Masako on her royal duties. Notice the wooden fan in her hand.