I know I used to see them at the very end of Hollywood movie credits. Why did they use them there?
Research paper outlines.
Elizabeth II, and other such “Nth person of this name.”
General use? Clock dials, prefatory page numbering, …, too many uses to enumerate.
Why do some movie credits use them in dates? A wild guess is that it’s classical, and anything classical is cool, following the principle of quidquid latine dictum sit altum videtur.
there is that famous guy Malcolm ten. 
Official or working titles of installments of a book or movie series (e.g., Star Wars: Episode IX; The Science of Discworld IV: Judgement Day)
Legal citations (e.g., Title 56, part III of the U.S. Code)
Numbering of Olympic games (“Games of the XXXII Olympiad” is the official name of the upcoming 2020 Summer Olympics in Tokyo)
I thought Cecil wrote a column about why copyright dates in movies are (or were) in Roman numerals. It was, as I recall, to obscure from the viewer that they were watching a re-release of an old movie instead of a new one. If Cecil did write such a column, I can’t find it now.
Movie sequel titles. Star Trek II, III IV V, VI. Rocky II, III, IV, V. etc.
The BBC still dates TV production years in Roman numerals: MMXIX
MMXIX is starting to look pretty good.
For a few years after MM - MMI, MMII… the symbols didn’t have much impact.
I expect they come back once we’re pat the boring bits.
World War I and World War II
Something in music theory called Roman numeral analysis
When discussing Deep Purple, the lineups are commonly referred to as Mk I thru Mk IX.
Many of these uses (part VI, mark IX, musical notes and similar) seem to be general enumeration of small numbers, so that seems to be a widespread current use, as opposed to something like “vii ounces” or any actual financial, scientific, or mathematical calculations.
This is the practice of symbolizing the chords in a key with Roman numerals.
There are seven notes in a common major or minor key/scale. Each note can be the basis of a (triad) chord.
So the first note in C major is C (natural). The basic triad based on that note is C-E-G. So C-E-G is notated as the I chord in C major. Similarly, in G Major, the I chord is G-B-D.
If you note the chords numerically this way, then you can easily change the key and leave the chord notations intact.
Next year’s Super Bowl should be dedicated to LIV Tyler.
Some companies use them on product names. The SONY RX100 series digital camera is up to RX100 VI now.
Of course there is the iPhone XR – the latest model.
I’ve assumed this was the “eks are” model until I recently saw a commercial where they pronounced it “ten are.” A very stupid naming decision in my opinion.
And you set the amplifier all the way up to XI
Those Chinese. They always have to go one better than us.
I saw it today on Todd Gurley’s jersey - GURLEY II.
Mind you he was at Super Bowl LIII.
Oddly, and for no legal reason, in the little copyright fine print in movies. Also in the designation of army corps. II Corps (and so on) is by convention pronounced “Two-Corps.”