Paperfolding naturally depends on the presence of paper. Thus we guess that the art is oldest in China and Egypt, but the oldest descriptions of origami we have are Mayan. However, it is in Japan that origami developed into a significant part of the tradition, in particular with ceremonial and pedagogical purpose as the amount of available paper after all was limited.
That changes during the 19-hundreds, and in the middel of the century enthusiasts all over the world explores and invents still more aspects, techniques, and models. We see people like Samuel Randlett, Gershon Legman, Adolpho Cercada, Ligia Montoya, and the greatest of them all, Akira Yoshisawa, who became a Japanese national icon. Yoshizawa develops the set of diagramming techniques and symbols which almost everybody follows, in variations, today and which implies that you can fold even the most complex models without knowing that language of the text.
Time is ripe for a wider knowledge of origami. With great enthusiasm the American Lillian Oppenheimer creates the first origami center, OrigamiUSA. She also introduces the Japanese term for paperfolding, origami. The word comes from "oru”, to fold, and "kami", paper.
The British magician Robert Harbin independently discovers origame and spreads it in England in the 1950's, and he later gets BBC to do a weekly origami show with him as entertainer, and he becomes co-founder of the British Origami Society.
In Denmark origami has been known to a lesser extent during the years. The first origami book, Folderier ("foldings") was published in 1944. Later, Thoki Yenn (1919-2004) became widely known for his origami. During the first many years he entertained with magic, paper cuts (kirigami) and films for children, but after meeting Lillian Oppenheimer in the late 50's he became also an enthusiastic paper folder.
He created the Danish Origami Center in 1991 where I experienced him as a very lively, intelligent and friendly man. Unfortunately the center folded when Thoki Yenn felt too old to manage the administration.
In the year 2006 the site papirfoldning.dk was created, and in 2010 a proper Danish origami society, foldning.dk was formed.
If you want to know more about the history of origami, take a look at the essays of David Lister which contains lots of interesting reading.