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I have studied Roman numeral analysis in my theory courses, and I think that it's a good way of describing and understanding classical music, especially that of the classical and early romantic periods.
I have heard from my colleagues that some theorists disagree. I've tried to find arguments online that Roman Numeral analysis is not useful, but I have not found any. What case (or cases) do theorists make against RNA?
The obvious shortcoming is that after we leave the classical period, music and tonality becomes too complex for Roman numeral analysis to be completely useful. So, we don't need to mention 9th chords or jazz 7ths and the like, and I believe you understand that already from the question.
For classical music, Roman numeral analysis is the most widely accepted method for describing common practice harmony, and I would agree that it is useful enough that every musician should be familiar with it. Furthermore, figured bass notation is highly related to Roman numeral notation in its more basic formats, and as such is important for musicians to learn especially if they are to be performing Baroque and early music.
However, the very obvious shortcoming of Roman Numeral analysis is that it analyzes music primarily in a vertical fashion, which is contrary to how we as listeners perceive music: as changes in sound over time, i.e. horizontally.
The works of J.S. Bach have, in many cases, eluded academic consensus on how they should be analyzed for centuries, and it is in the music of J.S. Bach that other methods of analysis become truly useful.
One such method is known as the Heptachord Shift. In short, rather than analyzing the harmony and function of each chord in isolation, this method tracks changes in the heptachord (that is, the set of seven pitches that defines musical tonality, or Do through Ti) over time by classifying notes outside of the current tonality to be alterations to the existing heptachord. In doing so, it aims to more accurately approximate how we experience music by analyzing change over time. In other words, a horizontal analysis.
The resource for this system of analysis is a 2001 paper by Marianne Ploger, who developed it, titled:
Heptachord Shift: A Real-Time Approach to Tracking Tonal Modulation Employing Precepts Observed in the Works of J.S. Bach. Marianne Ploger, 2001. (link)
(Your best bet on finding the article might be the University of Michigan master's thesis archives, or by emailing the writer directly.)
The only thing that I have found difficult to describe in Roman Numerals is 9th and higher chords in inversions. For example, a C9 chord with the G in the bass would be described in lead sheet notation as C9/G but my theory class never really came up with a consistent way to describe inversions beyond 7ths.
I sadly don't have any texts or examples beyond my own experience there. The downside to the lead sheet notation listed above is that it does not describe chordal relationships and harmony. :-P
I think you'll find pros and cons for any method -- and the more complex it is, the simpler it'll be to just use the notes!
Let me add my US $0.02.
Roman numeral analysis is very good for studying a piece of music and figuring out the harmonic progression.
However, if all you want to do is sight-read and play the music on your instrument, it's more efficient to name chords by their letter name because it's easier to read and play.
Let me give a very simple example. If you are presented with I - IV - V and asked to play it, you have to figure out the key center at that moment in the music, and which chord names correspond to those numbers. However, if you are presented with D major - G major - A major, you don't have to stop and think so much about what to play. The more complex the chords, as in jazz, and the more the music modulates from one key center to the next, the more that Roman numeral analysis gets in the way of performance. Borrowed chords (V7 of V or vii dim 7 of V, for example) would be hard for anybody to figure out what to play on sight-reading, but if you just label the chord by its root note and give a slash indicating the bottom note (indicating the inversion of the chord), it's easier to sight-read and to improvise on top of it.
I agree with much of the answers. For me they are great but yes it can get tricky with modern key changes, but if you think of modulations as phrases that arrive at a goal (unless abruptly jumping right into a new key with out prep) then thinking in terms of common harmonic patterns into the new goal key as I can be useful.
Let's say I'm in Cmajor and now I start a II-V-I pattern into Bb. But this is an analysis prep tool. Also using the flat/sharp signs in conjunction with romans ie.bVI7-V7-I etc helps to see how the chords are functioning chromatically. I think they're more objective and easier to keep track of when looking for musical patterns (Substitutions in jazz).
Also when I consider the function of voicings that I like by ear I can consider the notes over many different roman based patters and see how far out or in it really is. So I say use them but in conjunction with other symbol naming systems and mostly in analysis pattern observing/composition mode. I don't have much experience with atonal music. Sorry if my answer is too much in praise and jazz based.
Roman numeral analysis (RNA) has a variety of limitations — such as in analyzing non-Tonal music — as outlined in other answers. However, there is a specific and ongoing "controversy" among music theorists with regard to RNA.
Beginning in the mid 1980s, a number of influential theorists began constructing what is now known as neo-Riemannian theory, which provides an alternative to RNA for describing harmony and voice-leading. Neo-Riemannian theory developed in part to better describe late Romantic music (as opposed to Classical or early Romantic), which is to say, the highly chromatic music created at a time when composers were actively attempting to undermine the Tonal system (that RNA describes). The case for neo-Riemannian theory is that it better describes the harmonic transformations in highly chromatic music (say, Brahms or Wagner) than RNA. Although it can be (and has been) applied to earlier composers (particularly when they wrote highly chromatic music), its development has primarily focused on the late Romantic and later musics.
There can be heated debate between adherents of each theory as well as among those developing neo-Riemannian theory itself. This is what I suspect you and your colleagues were hearing about, as it continues to be a highly active area of research.
What is bad about Roman Numeral Analysis?
I don't actually recall hearing criticism of RNA, but there are things I dislike about it.
There are several, incompatible notation standards. E.g. wikipedia lists: conventional notation, alternative notation, chord symbols (https://en.wikipedia.org/wiki/Roman_numeral_analysis#Diatonic_scales) and Jazz/pop numerals (https://en.wikipedia.org/wiki/Roman_numeral_analysis#Jazz_and_pop_numerals). I can see people on Music SE getting confused if they follow different conventions on when to put accidentals or when to use capital or small letters.
It doesn't provide that much understanding of functional harmony. There are only 3 primary functions (T, S, D), but as many as 7 RN. The numerals don't tell anything about functions of chords other than I, IV and V. Why does ii-V-I sound good? (Could it be ii has something to do with IV?) Why V can resolve deceptively to vi? (Could it be vi and I have something in common?) You just have to accept it and learn by heart.
An advantage of RN is that many people recognize it, but it fails to be a consistent notation system or informative analysis tool.
The main criticism against Roman Numeral Analysis (RNA) regards its foundational premise. Although RNA appears to clarify the music by naming each chord within a key by its scale degree value (I, ii, IV, etc.); this does not actually amount to any understanding of how chords are related to each other or what the chords mean within a given key.
The lack of real meaning in RNA can be illustrated as follows. Why does ii-V-I sound good? (Could it be ii has something to do with IV?) Why can V resolve deceptively to vi? (Could it be vi and I have something in common and if they do how does the naming of vi and I show any type of relationship). Or consider this, why is vii related to V? If they are related shouldn't their numerical value be closer together or even the same? You just have to accept it and learn it by heart as in RNA there is no underlying logic which governs the relationship of chords or their logical progression.
Functional Analysis (FA) does however show how chords are related and offers how to make logical progressions. For example in FA, the I chord and the vi chord are termed Tonic (T) and Tonic Parallel (Tp) (note: parallel means relative in German theory). So immediately you can see how those two chords (I and vi) are very near in function because they are both given the base name T (one is T, the other is Tp). Take for example V and vii; in FA those are termed "Dominant" and "Dominant 7 with omitted root"; again you can see how they are shown to be near to each other as they both have the same Dominant base. Hugo Riemann is the main proponent of Functional Theory but his theory is so little understood in North American that many musicians are not even aware of this system.
