I vaguely remember descriptions of the two great heresies faced by the early Christian Church. These “heresies” or errors, were condemned by the early church fathers, and the adherents either banned or otherwise silenced. Anyway, are there still people who cling to these beliefs? And, if one were to revive such a heresy, what would the RC Church do to respond to it?
The Jehovah’s Witnesses have views that seem very like those of Arianism (“Christ is God’s Son and is inferior to Him”…“Christ was first of God’s creations”). Historically, I’ve never heard anything to suggest that the Jehovah’s Witnesses were some secret underground continuation of Arianism that survived through the millennia or anything Dan-Brown-worthy like that; I don’t know if Charles Taze Russell was familiar with accounts of the teachings of Arianism or came up with his ideas completely independently, or if modern Jehovah’s Witnesses would claim Arius as a forerunner of theirs, acknowledge any similarity of doctrines, or disclaim the resemblance as superficial.
I suspect that ideas which the theologically conservative would denounce as Pelagian might be pretty widespread among a lot of people who consider themselves Christians, without them necessarily having any familiarity with the term Pelagianism or the historical controversy over those ideas.
As to what the Catholic Church, or any other church would do about it–well, I expect they’d preach against it. It’s not like the Pope’s going to fire up the Inquisition or declare a Crusade or something like that. Battling heresy ain’t what it used to be.
(If you do some web searching, you will find a lot of evangelical Protestant web sites attacking the Jehovah’s Witnesses as being a non-Christian “cult”.)
They come up constantly or, more accurately, repeatedly. All those people who say “I respect Christ as a great teacher, but . . .” and then explicitly or implicitly deny his divinity are Arians. You could argue, I suppose, that modern secular Western society is in a way Arian, having accepting social and moral values that are derived from (or at least have been reinforced by) Christian teaching while denying the divinity of Christ. And you could also argue that it is Pelagian, asserting the ability of humanity to improve and redeem itself.
What would the Catholic church do? It does what any Christian church does - asserts Christian belief as it understands it, and urge people to accept it. Why should it react to Arianism or Pelagianism differently than any other kind of non-Christian belief?
Pelagianism does sound a lot like the later Protestant movements. Especially the belief in salvation without the church. So you could say that pelagistic beliefs are still around even if the official fan club went chapter 11.
The RC church can’t really do much about it in the US except try to push their own beliefs. The lack of religious education here in the US must really hamper that. Catholic Dopers: if you went to a catholic school, were these kind of things covered in your education?
You might be intrigued by this, a poll I started over on Christian Forums, in which I mixed two “orthodox” statements with eight statements describing early heresies, many of which got about as many votes as the “orthodox” ones, from people in churches which officially subscribed to the “orthodox” views regarding them as heretical.
I went to Catholic primary and secondary schools in the UK and had never heard of these concepts until today.
I didn’t go to Catholic school, but these ideas (not these terms) were taught. Salvation only with the church is pretty much what the RC is big on, afterall. Infact, it is in the creed along with the equality of each of the Trinity members. Again, I don’t recall those terms being used. Of course, in small town Catholic churches we had a hard time pronouncing the names in the readings, so adding any heretical terms or such would likely have been skipped right over. 
JW’s are indeed the closest to the Arians, and Charles Taze Russell did regard Arius as a forerunner.
Those who regard Jesus as just a great teacher or a “son of God even as we are are sons & daughters of God” are NOT Arians. Arius & Co. regarded JC as Father God’s first creation, the co-creator with the Father of everything else, Lord & Savior & greatest being of all creation. There are a few “Christian Unitarians” who may also hold to Arianism.
Pelagians are just radical free-willists. Arminius, Wesley & Alexander Campbell (the Restoration movement which produced the Disciples of Christ, the Churches of Christ, and the United Church of Christ) have been labeled as being semi-Pelagian. Some of the “Open Theism” proponents could be regarded as Pelagians.