The quartet is named for the 19th century violin makers Giovanni and Vincenzo Cavani.
The quartet has recorded music by Béla Bartók, Maurice Ravel, Antonín Dvořák, Robert Schumann, Johannes Brahms, Dan Welcher, Donald Erb, Ernest Chausson, Leo Brouwer, and James Primosch for the Albany, Azica, Gasparo, New World, and Pantheon labels.
For their 20th anniversary, they commissioned the work Musings for String Quintet from composer Joseph Hallman.
The quartet has been partners with many organizations, including the Chamber Music Connection, led by Debbie Price.
In June 2018, the Cleveland Institute of Music announced the severance of its affiliation with the Cavani String Quartet.