First off, the obligatory definition of terms: I'll be using "universalism" to mean "everyone will be saved and get to heaven eventually." Hopefully that's what Adam meant, or he's going to get a whole load of tangent.
The best place I know of to start when talking about Mormonism and universalism is the Vision--more commonly known today as D&C 76*. In Rough Stone Rolling, Richard Bushman discusses this universalist-esque revelation that sends such a tiny portion of people to hell as to almost be Universal in its scope. While Bushman notes--quite well, I think--that it is distinguished from universalist doctrines in important ways, it is also interesting to note the reaction from some members who thought it was too close to salvation for everyone: one member was excommunicated for proclaiming that "the vision was of the Devil came from hel[l]." This is a good reminder of how inclusive our basic doctrine is: virtually every single person ever to have lived on the earth will at least achieve an inheritance in a kingdom that "surpasses all understanding"--this is de facto universalism as far as most of Christianity (and the world) are concerned. So Adam, in a sense, Mormonism and universalism are very compatible, almost even identical. But I have the feeling that's not what you were asking, since that is basic Mormon doctrine that every Sunbeam knows.
Moving into speculation-land, then: can universalist doctrines apply to the really Mormon concepts of heaven? I mean, no Mormon is excited about making it to the Telestial kingdom, so can we reconcile universal salvation in the Celestial kingdom with Mormon doctrine? I propose that it is possible, though I don't make any guarantees that it's true.
There is an idea about the possibility of progressing from one kingdom of glory to another. It is controversial--Bruce R. McConkie called it one of his 7 Deadly Heresies (though he also called evolution one, and that is taught at BYU, so what does that say? :) He decried the idea that God is progressing too, which I also agree with) and Spencer W. Kimball, Joseph Fielding Smith, and George Albert Smith are also on record opposing it.
However... you also have James E. Talmage, Brigham Young (via Wilford Woodruff's journal), Joseph F. Smith, J. Reuben Clark, and B. H. Roberts in favor of it, with the Secretary to the First Presidency issuing a letter (twice) saying that the church has no official position on the question, plus Lorenzo Snow and Harold B. Lee saying things that seem to imply the possibility. (see here for most of the quotes (don't miss the first comment there as well for the Roberts quote) and here for Talmage's softening stance between editions of Articles of Faith).
My point with all these quotes isn't to prove the doctrine either true or false--there are pretty impressive people on both sides--but just to show that both are (I believe) valid options for believing Latter-day Saints.
Again, I don't claim to have a testimony about this issue, but as a generally merciful-leaning type I hope that there is progression available between kingdoms. True, the scriptures at first glance don't seem to support the idea (and even appear to debunk it), but I don't think it's quite that simple. D&C 19 makes it clear that sometimes God lets us believe things to be harsher than they really are so that we will be motivated to do what's right, which is actually beneficial to us. So yes, I do think it's possible that there will be progression between kingdoms and thus universal salvation in the Celestial kingdom. I wouldn't bet on it (a la Pascal's wager) but I hope for it. As it is, I highly encourage everyone to do what they can to get to the Celestial kingdom on the first go-round as it will make you happier sooner at the very least.
What about the sons of perdition? Can we create a true Mormon universalism? Well, D&C 76 describes outer darkness a lot like D&C 19 said hell is often described--that is, sounding like it's endless but really just meaning God-given (and thus very intense and long), so maybe even they'll get out eventually. As verses 44-46 say, only they who are consigned to that state will know the end thereof. As an alternative speculation, Brigham Young once voiced his opinion that the sons of perdition would be recycled (for lack of a better term) into their native element and get another chance at some kind of kingdom!
So Adam, does that answer your question? To sum up: Mormonism definitely includes saving virtually everyone in some version of heaven, and it's possible that it extends that to exaltation as well. Fascinating to think about, but important to remember that we should repent and work out our salvation as best we can today!
P.S. There was a great short story in a recent edition of Dialogue about a guy in the Terrestrial kingdom that touches on this subject. It's called "Eternal Misfit" by Roger Terry. I highly recommend it. If you're super interested, email me and I might be able to email you a pdf of it.
* As an aside, does anyone else hate the new online scripture format? I don't want to scroll through 75 section summaries to get to the link I want.