Both coffee and tea were introduced into common use in Western Europe during the later 1600s (17th century). Tea was originally imported from China and Southeast Asia, while coffee was introduced by the Ottomans, who borrowed it from the Arabs.

Many food plants that are now considered integral parts of the local cuisine in Africa, Asia, and Europe, such as potatoes, tomatoes, and chili peppers, are all originally developed in to the New World.

Unusually for a military technology, these some evidence that early rocket artilliery of the late 18th century may have been inspired by South Indian military rocket artillery designs. Before this Europeans usually used solid shot rather than exploding shells or self-propelled rockets. The National Aeronautical Laboratory of India published a lecture series on this back in 1985: http://www.nal.res.in/pdf/pdfrocket.pdf

Right up into early 18th century, Chinese techniques for the production of luxury ceramics remained superior in quality and cost to contemporary European attempts to duplicate it.

The whole world has been trading goods and ideas for a long as there have been people. Would you be willing to discuss the assumptions behind your question?

I posted this a year ago using the username /u/stal23, but it's relevant to this discussion:

In the West, Chinese philosophy and culture was very much in vogue among the intellectual elites of the Enlightenment. This fad of emulating China was called the "China cult." Chinoiserie, a style that borrowed Chinese motifs, was very fashionable in ceramics, painting, furniture, and textiles. François Quesnay was even able to convince Louis XV to perform a Chinese New Year spring plowing ritual. The trend was not without its critics, many of Europe's "sinophiles" were accused as propagators of Oriental despotism, an allegation that was to an extent not far off the mark. Voltaire, François Quesnay, and Gottfried Wilhelm Leibniz were three prominent figures that were a part of this trend (there were others, but I'll just cover three for now).

Voltaire: Voltaire's sinophilism was based on two aspects, religion and politics. He often used Confucianism as a platform to attack what he didn't like about Catholicism (and there was much he didn't like). The existence of Confucianism in his view was a powerful argument that the Church didn't have a monopoly on the moral universe. Voltaire admired what he interpreted as deism in Confucianism. This is technically not accurate, Confucianism is actually explicitly agnostic about God, but to Voltaire it represented a reification of his own beliefs in deistic monotheism, humanism, and religious tolerance. He was attracted to the political dimension of Confucianism. Chinese politics was, from his understanding, an enlightened monarchy. Voltaire dismissed critics that labeled it a form of despotism. His defense was that the system was based on benevolent absolutism, a paternal monarchy, led by what the Jesuits returning from the Chinese court had informed him was a "philosopher king" (no conflict of interest there!).

Leibniz: Like Voltaire, Leibniz was receptive towards Confucianism, but also towards the metaphysics of Taoism. He detected that there was a mathematical quality to Taoism, something which gelled well with his own understanding of mathematics and theology. Leibniz also believed that Chinese characters represented a characteristica universalis, an ideal language that could be understood universally because it was ideographic (it actually isn't). He wished to form his own characteristica, but gave up when he realized it was impossible to devise.

Quesnay: Quesnay was called the "Confucius of Europe", and for good reason. Quesnay and the Physiocrats believed an in agrarian-based economy, so a philosophy exported from the agricultural society of China fit neatly into their world view. And like Voltaire, Quesnay was a proponent of benevolent absolutism, a political system he thought was substantiated by the practice of Confucianism in the East.

Further reading: J.J. Clarke's Oriental Enlightenment: The Encounter Between Asian and Western Thought (1997)

"Voltaire, Sinophile" Arnold H. Rowbotham. PMLA. Vol. 47, No. 4 (Dec., 1932), pp. 1050-1065

Francois Quesney's Le Despotisme de la Chine (1767).

The aspect of East to West influence that always strikes me is the introduction of animals and foodstuffs.

Chickens, for example, are a jungle bird native to Vietnam, introduced into Anatolia in the ca. 400's B.C. Onions are from India, apples are from the Khazakstan / China region. Oranges were introduced to the West from Persia by Alexander the Great and his armies. Carrots seem to have been introduced into Europe far earlier (by at least ca. 3,000 - 2,000 B.C.).

But it all makes one wonder - how exciting was Western cuisine prior to all these introductions, and what interesting recipes have we lost out on?

Before the late 1700s, Europeans wore linen for their vegetable fabric, but then cotton was introduced from India. It not only took over women's clothes - all those cotton Directoire white gowns, later sprigged muslins and chintz patterns - but created the cotton industry in the American South that made slavery so attractive to plantation owners that they would go to war to preserve it.

Pajamas, as pyjamas, were brought from India about the same time. A loose trouser and button-up tunic, they were originally sportswear, worn for hunting. In Boucher, 20,000 Years of Fashion, they show a gentleman hunting in pink pajamas.