Musical Masses

So there is a long tradition of putting masses to music? Were the masses of Mozart, Bach, Beethoven composed to be listened to? or to actually be used during a Mass?? I have the Ave Maria Mass by Stephen Edwards and like it. It was commissioned by the great Thomas Monaghan for the founding of Ave Maria College and School of Law but then dedicated to the victims of Sept. 11th. The Kyrie is beautiful, so is the Sanctus. The Gratias Deo at the end is stirring and inspirational.