I've talked some about contact with Europe before here, but that is more about the fleeting contact they might have had and I actually find the answer somewhat flawed now. I think the biggest change would be my opinion on Pyrrhonism and the Indian influence on Pyrron. Since then I believe that Pyrrhon may have encountered the Ajñanas or possibly a proto-Madhyamaka Buddhist sect, in turn, it is likely his philosophy came back to India and helped influence Nāgārjuna. McEvily concludes that Sextus Empiricus' work came to India and influenced Nāgārjuna, but since they were roughly contemporary (Nāgārjuna was perhaps earlier), I don't think that is the case, he also discounts any influence of India on Pyrron, saying that the Greeks already had the tetralema since Democritus, but I think he goes too far in that, Pyrron's philosophy does closely resemble a variation of Greek Philosophy. It would be wrong to call Pyrron and Pyrronists Buddhists though. Their Ataxia is markedly different from Buddhist Nibbana despite comparisons certain authors have made.^1,2

There is also the case of Kalanos, an ascetic who went with Alexander and then committed suicide by fire after falling ill. Early scholars suggested he was a Jain, but more recently it has been posited that he was a Buddhist since the method is wrong for Jains ritual suicide may have been a part of early heterodox Buddhism and there may have been lone Buddhist ascetics who were closer to other groups.^3 Personally, that is a lot of maybes and possiblies so I would suggest that he was an Ājīvika due to their ritual suicides and association with fire, but this is just my personal speculation based on the evidence available to me.

In any case, Buddhism had next to no presence or understanding in the West until the 18th century. People at times were aware of Buddhism, but only through indirect transmission and hearsay. During the late 18th century and into the early 19th, considerable interest in the texts of Buddhism and Hinduism was developed among philosophers for their different perspectives from classical western philosophy. Numerous Journals and dictionaries were made and the Brittan, the Netherlands, Portugal, and France all began publishing journals and translations on Asia, it's history, and philosophy, much to the interest of historians and philosophers. Perhaps the most prominent of these philosophers in the early period was Friedrich Schlegel, who studied Sanskrit and was somewhat enamored with the works of Hinduism, especially the Bhagavad Gita. The Romantics were somewhat dismissive of Buddhism, but have in part shaped many of the misconceptions about it through influence of other philosophers such as Huxley.^4

The romantics had little impact outside of German scholars but the appreciation of Indian texts in Europe had a profound influence outside of them on scholars like the (in)famous Arthur Schopenhauer. Schopenhauer was perhaps the first champion of Buddhism in western philosophy and at various points called himself a Buddhist, or inspired by Buddhism. He was enamored with the ideas of suffering, logic, and emptiness. It was only after he began forming his original philosophy that he discovered Buddhism. Initially, it was the Upaniṣads that captured his interest, forming a trinity with Kant and Plato as the great works of Philosophy. When he discovered Buddhism he was shocked but also quite happy at the similarities and was compelled to include it into his magnum opus The World as Will and Sufficient Reason, though it first appeared in The Fourfold Root of Sufficient Reason. His later years were spent in solitary hermitage, seeking his interpretation of awakening.

Schopenhauer's philosophy was markedly different from Buddhism, and was notably more pessimistic. He did not believe in rebirth, and ridiculed some of the texts that professed the doctrine. Despite, Schopenhauer's misunderstanding of the doctrine, his opinions on Buddhism and Buddha were somewhat influential and along with the German Romantics have colored western and philosophers in their critiques of Buddhism.^5,6

While Shopenhauer's pessimistic and difficult philosophy and texts were not widely circulated outside of the intelligentsia, it was Sir Edwin Arnold's The Light of Asia which brought Buddhism to the mainstream awareness to Europe. While Edwin Arnold was not a Buddhist himself, unlike Schopenhauer he never professed it at least, he was an Academic dedicated to bringing Indian philosophy and literature to the west and raising awareness of it. The Light of Asia was an incredible success and was published in many languages, somewhat unusual for a poem of the era. Edwin Arnold was particularly passionate about Buddhism and was strongly dedicated to spreading the texts such as his later publication of The Essence of Buddhism a pamphlet sized selection of Buddhist texts. Later, he would be a founding member of the Maha Bodhi Society, though that would only happen later.

Shortly after this, Henry Steel Olcott published Buddhist Catechisms a distillation of the doctrine as understood by him from his time in Sri Lanka featuring selections from the Pali Canon. Olcott would go onto form the Theosophical society along with Helena Blavatsky. The Theosophical Society was a syncretic fusion of Indian philosophies with Christian mysticism that professed itself as a new doctrine for the coming era. They saw Buddhism as one of the primary foundations for their new philosophy along with Occultism, Gnosticism, Esoteric Christianity, and Vedanta. The Theosophists weren't (and still aren't) Buddhists. Instead they saw all religions as a part of a greater whole and the path to the ultimate goal. That said, the founders all publically took refuge in the Dharma.^7

Regardless of the Theosophists actual beliefs, they were instrumental in the revival of Buddhism in India and Sri Lanka as well as the spreading of Buddhism to the west, though in a somewhat distorted form. On top of this, the Maha Bodhi Society established a center in America and in Germany, the first Germans went east to become ordained and founded societies for the promotion of Buddhism in Europe. World War I ended the burgeoning Buddhist movements and stalled missionary efforts, the desolation of the war left Europe in pieces, but following the end the second wave of Buddhist movement began.

This second wave took place mainly in Germany and England, mainly in the same vein as the early wave. The players were the same as before, led by the Maha Bodhi Society and Theosophical Society. The Theosophical Society actually did become more Buddhist and Vedantan over time, especially the Maitreya Project, which took the name of the Future Buddha and prophesied the coming of a "World Teacher", what happened there is a whole different story.

Like before, these efforts were squashed by the rise of Nazi Germany (most of the early converts to Buddhism were German) and subsequently World War II, which also isolated Japan from the west. However, following World War II interest in Buddhism exploded. D.T. Suzuki became an international celebrity and Zen temples were established worldwide. Tibetan Buddhism became popular as well. Sokka Gakkai Also became a powerful force in the westward movement of Buddhism, with thousands of members abroad after WWII. From Zen Buddhism and the writings of D.T. Suzuki came the spirituality of the Beat Movement. People like Kerouac and Ginsberg became enamored with the philosophy of Zen and began to include it in their writings and attended lectures and retreats on Zen, on meditation, and on various esoteric practices. When we think of Zen in the west, typically we think of this "Beat Zen" led by Ginsberg and Kerouac who became the westward missionaries of Zen.

The ravages of unrest in Asia also sent thousands of Buddhists abroad and brought disparate traditions to Europe and America, such as Pure Land and Tibetan Buddhism. The self-immolation of Thich Quang Duc and the early writings of Thich Naht Hanh in France brought Vietnamese Buddhism to attention. Chogyam Trungpa's missions into America attracted the Beat movement leaders and academic leaders worldwide and brought in some of the converts to Buddhism.

The efforts of Academics has been important to the propegation of Buddhism as well. People like Richard Gombrich and the members of the Pali Text Society have always been important as well.