By special request, a short treatment of one of the major heresies follows.
It's a little difficult to put a name to this heresy, or rather this group of heresies. It is often called Gnosticism; often, more misleadingly, called Manichaeism, after its fifth-century incarnation. It is a perennial tendency of the human mind when it turns to spiritual things; indeed, though other such repeated themes exist (e.g. what might be called the "Third Age" heresies in which the revelation of Jesus Christ that superseded the Torah is itself superseded by something else, such as we see in Montanism, Mormonism, and arguably in Islam), this recurring heresy is the most persistent. It might, after a fashion, be called the heresy of spirituality.
The Catholic faith can be regarded as essentially a commentary upon three central mysteries (neither confusing the mysteries nor dividing the substance): the Trinity, the Incarnation, and the Redemption. Every doctrine, whether naturally supported or supernaturally revealed, exists in relation to these three doctrines; and the Incarnation is the linchpin, for it is the bridge between the other two, the means of revelation of the Trinity and the means of effecting the Redemption. The heresy of spirituality is an attack on, or more commonly a retreat from, the fullness of the Incarnation.
This to be expected. The human mind, when once it has conceived of a transcendent rather than a substantially anthropomorphic Deity, is offended and shocked by the notion of such a being coinhering with the smallness, the dirt, and the need of humanity. When once we begin dealing with spiritual things at all, it is easier to deal with them by themselves and disdain the physical, rather than adopting the Catholic sacramental mindset* -- as, when a man has been civilized and educated, he finds it easier to be snob than a gentleman. It is an easy reality to forget, because our own day and age has been (until quite recently) so shaped by the doctrines, and still more shaped by the imagery, of Christianity that it seems to us as if that is the natural form that religion takes. In truth, judged by the standards of literally every other religion that has ever existed, it is far more normal to have either a religion that is idealistic and regards the body as a hindrance or at best a thing of indifference, or a religion that is orgiastic, magical, and unreflective. The stern, clean spirit of the Stoics and the divine madness of the cults of Dionysus are the directions in which the human soul naturally goes; the sacramental, incarnational atmosphere of Catholic Christianity was something really new, and continues to be a shock to those outside of, if they can be made to understand it. The outrage of a Moslem or the bafflement of a Buddhist at being told that a personal God became Man is much more rational and natural than our humdrum acquiescence or humdrum rejection.
I have a sort of educated guess (which is half gut feeling) that we are in for a major revival of Gnostic thought. In part, its general absence from our culture for such a long period strikes me as fishy; but the extreme license of our age tends to point to it as well, I think. When people have seen everything, and done most of it, a certain amount of ennui is bound to set it. It is, thanks be to God, pretty hard to get human beings to be bored of sex (though, speaking as someone who has studied Classics, it is in fact possible); but when once that does happen, Gnosticism is one of the natural reactions. Not a desirable reaction, but quite an understandable one. This is a very good reason to emphasize not only the goodness but the true character of Catholic asceticism: it is not a rejection of the material world but a renunciation of its use -- a confession, "This created thing is very good, and it is not for me." It is, equally, a reason to emphasize the highly contrasted but naturally allied phenomenon in the faith of a boisterous indulgence in bodily pleasures: one that does not overstep the bounds of virtue, but that does rejoice in the pleasures and in the body as good things, made by God for enjoyment among other purposes. (Think of Irish Catholicism, at least as conceived of by Americans, and you've got the idea here.) Both true asceticism and what we might call a sanctified sensuality are replies to both modern licentiousness and the overreactive disgust with licentiousness that will likely take its place.
It is also a good reason to emphasize devotion to the Mother of God. Mary, in her function as Theotokos, symbolizes the union of affirmation and renunciation: motherhood and virginity being both realities in themselves, and images of all affirmation and all renunciation; neither immoral license nor any attack upon creation can be sustained alongside acknowledgment of her. She, and especially her title Mother of God, are a unique and irreplaceable assertion of the Incarnation of God.
Throughout the history of the Church, this heresy has taken various forms. There are hints of such thought being combatted in the New Testament itself, probably syntheses by Greek believers of philosophical ideas with the Catholic creed -- "spiritual" interpretations of things like the Resurrection, for instance, which St. Paul addressed in I Corinthians 15. But for the most part these sects do not seem to have been fully formulated during the lifetime of the Apostles.
In the first two or three centuries, there was a brace of heresies of this kind, loosely called by the collective name of Gnosticism. They were so called from the Greek term gnosis, meaning "knowledge," because they believed that salvation from this corrupt world came through knowledge -- in contradistinction to the Christian view that salvation from sin came through faith. The famous Nag Hammadi library, found in Egypt in 1945, was to Gnosticism what the Dead Sea Scrolls were to Judaism and Christianity; most of the "secret gospels" and so forth much touted in the media today, like the Gospel of Judas, were of this character. The Gnostics had a number of typical marks, such as secret passwords, an inner ring of "perfect" surrounded by an outer ring of camp-followers, complex genealogies of deities and angels, and so forth; but their essential character was twofold: salvation through knowledge, and a belief that spirit was good and matter was bad. The early bishops and apologists of the Church had no tolerance for this doctrine, which would have made Jesus an illusion (or in some systems, such as that of the Mandaeans, an impostor); St. Irenaeus and St. Ignatius of Antioch wrote forcefully and eloquently against the Gnostics.
Manichaeism -- often classified as a form of Gnosticism -- swept over the Roman Empire in the third, fourth, and fifth centuries. It combined Christian and Gnostic ideas to a certain extent (as had many Gnostic sects, such as the Marcionists). St. Augustine spent some time as a Manichee; it seemed to him to have a solution to the problem of evil, for, originating in Persia, it had borrowed the Zoroastrian idea of two eternal deities, one good and one evil, and assigned the responsibility for all bad things -- including the creation of the material world -- to the latter. It also indulged itself in some more puzzling and, if you're in an uncharitable mood, entertaining beliefs, as that cucumbers and melons contain imprisoned particles of divine light that are released when you eat them -- but the inner circle, who claimed they could hear the light particles screaming when these fruits were harvested, got believers of the outer circle to do the harvesting for them. St. Augustine converted first to Neoplatonism and then to Christianity, both of which discountenanced and to some extent despised Manichaeism, and wrote extensively against it.
The name Manichaeism stuck throughout the Middle Ages and later still -- possibly in part because of St. Augustine's influence, even more so because the Medieval mind was formed in the twilight of the classical Empire. For this reason, though they were also known as Cathars (from the Greek term for "pure") or Albigenses (from the French city of Albi, a major center of the heresy), the Medieval adherents of this type of belief were often called Manichees. They attacked all seven Catholic sacraments, the Eucharist particularly, and opposed more or less all sexuality: not only did they reject marriage, they normally refused all food that was the product of sexual reproduction -- meat, but also eggs and dairy. They rejected every kind of violence (in theory, anyway), a significant departure from the typical worldview of the Medievals, to put it mildly. They also rejected penance: someone who sinned after receiving what they called the consolamentum, an elevation to the status of a perfect believer, was irredeemable. Some went as far as to starve themselves after receiving the consolamentum, to hasten death. The sect lasted only a few centuries; between the Inquisition, the Albigensian Crusade (proclaimed in 1208), and the missionary activity of the Dominican and Franciscan Orders, the Cathars had more or less ceased to exist by the middle of the fourteenth century.
Between the Late Middle Ages (roughly 1300-1450) and the end of the nineteenth century, Gnosticism as such does not seem to have reared its head much. It is possible that it was preserved in certain secret societies, such as the Rosicrucians; but these often had an alchemical or Christian coloring that prevented a fully Gnostic attitude toward the material world. Many Catholic thinkers (and even some Anglican and Lutheran ones) have identified Calvinism as the heir of the Gnostics during this period. It cannot be denied that there are similarities, especially in the Calvinist emphasis upon predestination. In some ways, that similarity has increased as time has gone on: Calvin and his immediate successors, for example, were sacramentalists, in the tradition of Augustine -- that is, they believed the sacraments really conveyed what they signify, although Calvin interpreted this differently than the Catholic Church does; but modern Calvinists typically don't believe that the sacraments actually convey grace to the believer (at most, they are held to be occasions of grace). The popular objection to Purgatory is of a similarly Gnostic character -- as a friend once put it to me, "Once we're free of the sinful flesh, we don't need to be purified any more"; though to do them justice, many Calvinists, while fully disbelieving in Purgatory, would reject that reasoning forcibly. I'm not satisfied that Calvinism can be regarded as having more than a partly superficial and partly coincidental resemblance to the spiritual heresy: although it mostly lacks the doctrine of the sacraments and the practice of the liturgy that characterize Catholicism and its ideological relatives, and although it tends to be hostile to Mariology, it confesses the Incarnation in quite as orthodox a fashion as the Council of Chalcedon.
As the Victorian era drew toward its close, however, this Gnostic trend of thinking experienced a revival. The Theosophical Society of Madame Blavatsky was one example, a pantheistic mystical group; Spiritualism was another, which often viewed a disembodied afterlife as a higher plane of existence (in marked contrast to the Christian expectancy of final resurrection). During and since that period, down to the present day, a number of formally and explicitly Gnostic churches, claiming to revive the ancient doctrines and practices, have come into being; others, such as the brilliant and depraved Aleister Crowley, have been content to draw on such ideas and radically alter them (some forms of Wicca, which, being a nature-oriented religion, is pretty un-Gnostic, owe something to these streams of thought, especially with regard to sex magic). Scientology has been noted by some thinkers to have pronouncedly Gnostic elements as well.
*This mindset is not exclusive to Catholicism: our Orthodox brethren have it quite as much as we do, and some Protestant bodies (chiefly of Anglican and Lutheran cast) have it as well. However, it is not a special mark of the evangelical world, whose doctrines and history have developed in an exceedingly different direction, centering on the Redemption. The sacramental worldview is not necessarily incompatible with Protestant doctrine per se; but the atmosphere is so unmistakably divergent that it is, in this country, meaningful to speak of it as specifically Catholic.