For no particular reason, I have put together a list of my favorite "theme and variations" classical pieces, with YouTube links and comments, neither especially knowledgable nor profound. Roughly descending order of preference.
I'm not much of a musical connoisseur when it comes to fine points of interpretation or performance, so I'm not going to argue that the particular performances I've linked here are the best. I enjoy them.
Bach: Chaconne in D Minor . The ne plus ultra of the form. There is a greater dramatic sweep and emotional intensity than in many Romantic pieces famous for those, and the formal structure feeds into that. Jascha Heifetz, violin
Like a lot of Bach, this works on many different instruments. Haym Hirsh and I put together a list of YouTube videos with 35 different instrumetations.
Schubert, Wanderer Fantasy. Not strict form, except in the second movement, but definitely overall largely a theme and variation. Again a huge dramatic structure. Performance by Evgeniy Kissin There's also a very interesting analysis by TRO. In particular he discusses the complexity of the harmonic structure, which I had never noticed.
Schubert, Death and the Maiden Quartet, second movement. The most intensely sad music I know. The final variation makes my heart race. Performance by Alban Berg Quartet
Mendelssohn, Variations sérieuses. If I were going to give a lecture on the classical "theme and variations" form, this is the one I would choose. There's a theme that is beautiful in itself, melancholy and harmonically complex. Then there's a diverse, imaginative collection of 17 variations that maintain the theme, then a fiery coda. What more could you want? Performance by Vladimir Horowitz. I'm not usually a great fan of organ music but this organ transcription by Reitze Smits, played by Johannes Gefferts works extremely well, it seems to me.
Mozart, Clarinet quintet, fourth movement. The clarinet quintet is one of my favorite Mozart pieces, and the last movement is perfect. When you have multiple instruments, different variations can correspond to different interactions among the instruments — Mozart does that brilliantly here. Performance by Joan Tormo, Laura Delgado, Javier Baltar, Sandra Garcia, Laura Szabo.
Beethoven, Kreutzer sonata, second movement. The Kreutzer sonata is one of my favorite Beethoven pieces. The beautiful second movement is very cheering after the intense first movement. Performance by Yuja Wang, piano; Joshua Bell, violin,
Mozart, Piano Concerto 24, third movement. The only orchestral piece in this list. Again, the variation form fits amazingly with the varying form of dialogue between piano and orchestra. There is also a great dramatic form, with the extended second to last variation followed by the danse macabre of the coda in 6/8 time. Performance by Mitsuko Uchida with Jeffrey Tate, English Chamber Orch.
Haydn, Andante with variations in F minor. Unusual form, because it has two alternating long themes, the first 41 measures in minor, the second 20 measures in major, and then there are only two variations plus coda. My favorite Haydn piano piece. Performance by Alfred Brendel. Discussion by András Schiff.
Beethoven, Appassionata sonata, second movement. Performance by Anna Fedorova
Schumann, Sonata in G minor, 2nd Movement. Melancholy and beautiful. Not in strict form. I had known this for years before it occurred to me that it is (essentially) a theme and variations. Performance by Martha Argerich. The third movement of Sonata #3, "Variations on a Theme by Clara Wieck" is also extraodinary, though I don't love it as much. Performance by Horowitz
Bach, Aria Variata alla maniera Italiana Like the Mendelssohn, an unusually rich and complex theme. This deserves to be better known than it is (unlike anything else on this list). Performance by Tatiana Kachko
Beethoven, Piano Sonata #30, Op. 109, third movement. Performance by András Schiff Performance by Elisabeth Brauss
Handel, Harmonious Blacksmith Performance by Wilhelm Kempff (Thanks to my sister Abby for pointing it out.)
Brahms, Variations on an original theme, Op. 21 No. 1 Performance by Arcadi Volodos
Tchaikovsky, Trio Op. 50, second movement. Huge, emotionally and musically. Performance by Yuja Wang, Leonidas Kavakos, and Gautier Capuçon
Beethoven, Piano Sonata Op. 26, first movement Performance by Sviatoslav Richter
Mozart, Piano Sonata 11 in A major, first movement Performance by Maria João Pires
Mozart, Piano Sonata 6 in D major, third movement Performance by Mitsuko Uchida
Bach, Goldberg variations. I mean, it's fine, it's a pleasant enough way to spend an hour. But I find the aria thin — I don't at all understand why people play it stand-alone as an encore — the variations not that interesting, and there's not much overall structure. I don't find that it has either the emotional power or even the musical depth of Bach at his best, though I can't be specific about what I find missing, and I certainly can't defend that claim. It's not one of my top 20 favorite Bach keyboard pieces.
Beethoven, Diabelli variations. Dull, dull, dull. Beethoven has nothing to say, and goes on for an hour not saying it. I once tried to force myself to listen to the whole thing and gave up half way through.
Beethoven, Sonata 32, Op. 111, second movement. People speak about this in hushed tones because it's the last movement of Beethoven's last piano sonata, though of course there's not much reason to suppose that he had that in mind. It doesn't do much for me. It seems to me — no doubt, if you were reserving judgment on whether I actually am a Philistine, this will clinch the deal — anyway, it seems to me that sometimes Beethoven has the idea that developing a series of variations consists in sticking more and more notes in. There's some of that in the Appassionata and in Op. 109, but there's enough else going on that they work. In Op. 111, there isn't enough else going on.
Mozart variations on Twinkle, twinkle, little star (Ah, vous dirais-je Maman). On YouTube you can find all kinds of videos of this or that banal tune (Happy Birthday, My Country 'Tis Of Thee etc.) as interpreted by various composers. These are generally (not always) competent, mildly amusing, generally superficial. These variations remind me of that. This is how Mozart would have written variations on Twinkle, twinkle, little star, except that, somewhat regrettably, he actually did. It's how people who dislike Mozart think of Mozart: pretty-pretty and shallow.
Chopin variations on La ci darem la mano. If this were some third-tier contemporary of Chopin — Moscheles, say — I would say, "Well, that's rather nice; occasionally it almost sounds like Chopin." But really, for Chopin?
Paganini Caprice 24, whether in his own version, Brahms', Rachmaninov's, Lutoslawski's, anybody's. . Yawn.