Abstract: It is easily verified that, if X, Y , Z are 2 x 2-matrices, then the following identity holds: (X Y - Y X)^2 Z - Z (X Y - Y X)^2 = 0. This is a simple non-trivial example of a polynomial identity. This identity appeared in 1937 in a paper by W. Wagner on the foundations of projective geometry. Are there other polynomial identities for matrices? How can we find more? What are they good for? I'll try to answer such questions in my talk.