Nowadays, soloists in a concerto play from memory, especially pianists. Occasionally, players of other instruments will use written music, but I have only seen one pianist using music. He was on the faculty when I was in graduate school, and students discussed the oddity for days afterward.
Since the piano requires the use of both hands, memorizing music for performances has the obvious benefit of not requiring a third hand to turn pages. It may come as a surprise, then, to learn that pianists have not always performed from memory and audiences have not always expected them to.
For a performance of his C minor piano concerto, Beethoven asked his friend Ignaz Ritter von Seyfried to turn pages for him. Seyfried later reported that the pages contained mostly blank staves, with a few cryptic notes here and there that he compared to Egyptian hieroglyphics.
At the time of the performance, Beethoven had not yet had time to write out the solo part, so he played it from memory. He apparently didn’t want the audience to know that he had not prepared his own part, thus his request to Seyfried.
Beethoven gave a signal when to turn the pages. Poor nervous Seyfried stared at them as if the whole notation had been present but hard to read, much to Beethoven’s amusement.