Ok, in formal writing, when we refer to a particular century adjectivally (i.e. seventeenth-century art) we use a hyphen. Now, what are we supposed to do with the twenty-first century? I’m having Twenty-first-century anxieties. Twentyfirst-century anxieties? Aiee! In my field, we use Turabian and it doesn’t mention this. Does anyone’s MLA or Chicago manual cover this? I’ve just hit my first occurance of this and it’s not going to get any better from here on out-- the century won’t be unhyphenated again until the thirtieth, so we should all work this out.
Chicago prescribes “twenty-first-century.” It’s quite logical and sensible, and multiple-hyphen adjectives are far from uncommon. Don’t sweat it.
The Chicago Manual of Style suggests that you are having twenty-first-century anxieties. 
(Note to self: refresh more often…) Sorry, Scarlett67.
Ok, it IS logical and sensible, but I was under the impression that multiple-comma adjectives were considered Satan’s work, and that you should try to get rid of the second comma or something (my advisor recently derided me for “late-fifteenth-century painting” but wasn’t quite sure what I was supposed to do with it). I’ll just try to calm down, then.
The simple solution is “21st-century.” For those who insist on stuffiness, “twenty-first-century” would work as the adjective. For your other example, I probably would have gone with “late fifteen-century” since “late” is modifying “fifteen-century.”
The goal is to be clear.
You hyphenate words working together as a single adjective to indicate that they belong together and do not separately modify the noun. An example of a possibly confusing construction would be “shocking pink dress,” which could be taken two ways. The dress may be shocking in its appearance and pink in color, or the dress may be shocking pink in color. If the latter is the case, you should write “shocking-pink dress” to eliminate the confusion. If the former is the case, you should rewrite the phrase. 
With a construction like “twenty-first century anxiety” there is precious little room for confusion, so I wouldn’t worry about it.
I’d write, “shocking, pink dress.”
I concur with KneadToKnow. Lose the second hyphen—it will look better and you won’t lose any clarity of meaning. In fact, I would have thought the Chicago Manual of Style would be the first to give this advice.
There’s a little club in an online community of copyeditors I belong to. We call ourselves HARPys --Hyphens Are the Reader’s Pal. Sure, there’s little room for confusion, but the second hyphen clearly binds the term together as a unit and eliminates the possibility of the phrase being misread even slightly, forcing the reader to have to stop and go back to get the precise meaning. “Twenty-first-century” as an adjective is supported by the Chicago, APA, MLA, and ACS style guides, as well as by Words Into Type. There’s no good reason to leave the second hyphen out.
What do you do about ambiguities that aren’t resolved by a hyphen? The Lion King was billed as “Disney’s thirty-second animated feature”. I figured “Yeah, it was short, but not that short”. OK, so that one was a bit silly, but it was a double-take, at least, and one could think of less silly examples.
What does the following mean?
“I wrote my twenty-first century synopsis.”
That would certainly be one option. The comma indicates unambiguously that both adjectives are equally modifying the noun. Personally, I’d rewrite it differently, but, hey, to each his or her own.
I certainly can’t argue with that. I often err on the side of being too clear, as anyone who’s read one of my 150-word “me-too” posts can tell you. (BTW, notice the hypenation :))
That is a prime example of bad writing and one of the best reasons I can think of to use numerals instead of writing out a number. “Disney’s 32nd animated feature” would have been the better choice, IMHO.
Here we have a good example of when you do need the hypen to be clear. It seems to me the choices are to add a hyphen between “first” and “century” or to rewrite the phrase. Personally, I’d rewrite the phrase, but again, that’s me.
Oooh, finally a chance to use all the authority of my new AP style guide. To quote:
“CENTURY Lowercase, spelling out numbers less than 10: the first century; the 20th century.”
So, if you want to describe your mental state to the newspaper readers of the world, you should be having “21st century anxieties.”
just my two centuries worth
[Uh, sorry. Guess I’ve been reading Snopes ‘no enlightenment without a bad pun’ web site too much]
I don’t mean to turn this into a debate, but I fail to see why “Disney’s thirty-second animated feature” is unclear. “Thirty-second” is an adjective modifying the noun “feature.” Written as “Disney’s thirty second animated feature,” “thirty” is an adjective modifying the noun “second.”
Also, I don’t see why you would rewrite “I wrote my twenty-first century synopis.” I don’t believe you could reword it without undue loquacity. I used it just to show that in fact, leaving out the second hyphen does indeed lose clarity (“clarity of meaning” seems redundant). As it stands, the statement means what it says, “This is the twenty-first synopsis I have written about the century.” The simple addition of the second hyphen makes it mean something else.
Yabbut, AP is newspaper style, and its main interest is conserving space and easing line breaks. AP uses numerals for ALL numbers, cardinal and ordinal, 10 and up – so the rule givan above is not peculiar to centuries. I wouldn’t recommend AP style as a guide for formal writing outside the journalism field.