The long-lived Saint-Saëns (1835 - 1921) is in the second generation of French Romantic composers after Hector Berlioz. As a young man he was influenced by the first generation of Romantics, Schumann and Liszt, and the later Wagner. But, like most French musicians of the time, he adhered to a more conservative classicism in his own career. He was a church organist and, for a few years, professor of music, during which he taught both Ravel and Fauré.
Saint-Saëns' cello concerto shows the influence of both Robert Schumann's piano concerto and the cyclic technique of Franz Liszt. Here is the opening:
As you can see, instead of an opening orchestral statement of the theme or themes (typical of 18th century concertos), after an opening forte chord in the orchestra, the cello begins with a virtuoso flourish, which turns out to be the first theme. The model for this is the Schumann Piano Concerto which influenced most 19th century concerto composers.
One innovation is that the concerto, while in the three traditional sections, fast-slow-fast, fuses these into one (20 minute) movement with no pauses. The last section, after recapitulating material from the first movement or section, ends with a new theme for the cello. The second movement or section is a minuet in the unusual key of B flat (with excursions into G minor), the Neapolitan!
Incidentally, Mozart occasionally inserted a minuet into the last movement of a piano concerto. You could consider this Saint-Saëns concerto as single movement with an interpolated minuet. In any case, it is somewhat innovative, though with classical precedent and, above all, very charming.
Let's have a listen. Here is a live performance by the great Mstislav Rostropovich. The London Philharmonic is conduced by Carlo Maria Giulini: