Hesychasm is foundational to Eastern Orthodox Christianity and is one of the things that distinguishes it from Roman Catholicism and the Protestant confessions that arose out of Roman Catholicism.
The watershed event in Orthodoxy regarding hesychasm was the so-called "Hesychastic Controversy", that occurred on Mt. Athos in the early 14th century. Gregory Palamas, a monk originally from Constantinople, helped define the Orthodox tradition as a result of the controversy.
The Orthodoxwiki entry on Gregory Palamas describes the event as follows:
[Gregory] He was initially asked by his fellow monks on Mount Athos to defend them from the charges of Barlaam. Barlaam believed that philosophers had better knowledge of God than did the prophets, and he valued education and learning more than contemplative prayer. He stated the unknowability of God in an extreme form, having been influenced by a reductionist interpretation of the writings of St. Dionysius the Areopagite. As such, he believed the monks on Mount Athos were wasting their time in contemplative prayer when they should instead be studying to gain intellectual knowledge.
When St. Gregory criticized Barlaam's rationalism, Barlaam replied with a vicious attack on the hesychastic life of the Athonite monks. Gregory's rebuttal was the Triads in defense of the Holy Hesychasts (c. 1338), a brilliant work whose teaching was affirmed by his fellow Hagiorites, who met together in a council during 1340-1341, issuing a statement known as the Hagioritic Tome, which supported Gregory's theology.
A synod held in Constantinople in 1341 also supported St. Gregory's views, condemning Barlaam. Later, in 1344, the opponents of hesychasm secured a condemnation for heresy and excommunication for Gregory, the saint's theology was reaffirmed at two further synods held in Constantinople in 1347 and 1351. Collectively, these three synods in Constantinople are held by many Orthodox Christians and several prominent theologians to constitute the Ninth Ecumenical Council. Between the latter two synods, Gregory composed the One Hundred and Fifty Chapters, a concise exposition of his theology.
This latter work - One Hundred and Fifty Chapters - can be found in English translation in Volume 4 of the Faber and Faber edition of The Philokalia.
The Orthodoxwiki entry gives some additional background on Gregory's hesychastic theology:
Contrary to Barlaam, Gregory asserted that the prophets in fact had greater knowledge of God, because they had actually seen or heard God himself. Addressing the question of how it is possible for humans to have knowledge of a transcendent and unknowable God, he drew a distinction between knowing God in his essence (in Greek, ουσία) and knowing God in his energies (in Greek, ενέργειαι). He maintained the Orthodox doctrine that it remains impossible to know God in his essence (God in himself), but possible to know God in his energies (to know what God does, and who he is in relation to the creation and to man), as God reveals himself to humanity. In doing so, he made reference to the Cappadocian Fathers [Basil the Great, Gregory of Nyssa, and Gregory of Nazianzus] and other early Christian writers.
Gregory further asserted that when the Apostles Peter, James and John witnessed the Transfiguration of Jesus Christ on Mount Tabor, that they were in fact seeing the uncreated light of God; and that it is possible for others to be granted to see that same uncreated light of God with the help of repentance, spiritual discipline and contemplative prayer, although not in any automatic or mechanistic fashion.
He continually stressed the Biblical vision of the human person as a united whole, both body and soul. Thus, he argued that the physical side of hesychastic prayer was an integral part of the contemplative monastic way, and that the claim by some of the monks of seeing the uncreated light was indeed legitimate. Like St. Simeon the New Theologian, he also laid great stress in his spiritual teaching on the vision of the divine light.
Gregory Palamas is such an important figure in Eastern Orthodoxy that the 2nd Sunday of Great Lent every year is reserved for his commemoration.