The exact origin of the art of origami is unknown.
Paper was invented in China in the first century A.D. and brought to Japan by Buddhist monks in the sixth century A.D.
However, the written records for the period are limited, so it is unknown whether origami first started in China or Japan.
However, no one will dispute that Japan developed origami to a very high art form.
Most origami instructions were passed on by oral tradition.
The oldest known written document about Japanese origami, the Senbazuru Orikata ("How to Fold One Thousand Cranes"), surfaced in 1797.
The first works of original modern origami (in the 1950's) are due to the master Yoshizawa Akira.
It is also known that the Arab world was making paper in the eight century, and the Moors brought paper folding to Spain in the twelfth century.
Paper folding or papiroflexia subsequently became very popular in Spain and South America.
The kindergarten movement in Germany (introduced around 1835 by Friedrich Froebel) included paper folding.
Paper folding was a popular children's hobby in the England of Queen Victoria, as evidenced by a couple of John Tunnel’s illustrations for Through the Looking Glass by Lewis Carroll, featuring two simple origami hats.
While origami is often considered a children's pastime in the West, its long history in Japan has enmeshed it in the cultural identity.
References:
1) Origami: A Brief History of the Ancient Art of Paperfolding, by Joseph Wu.
2) Origami Origins (from the General Libraries at the University of Texas at Austin)