If A and B are self-adjoint matrices and $A leq B$,

what functions $f$ have the property that $f(A)$ must be less than

or equal to $f(B)$?

This question was answered by K. Lowner in 1934.

Recently J. Agler, N. Young and I have extended this result to functions

of more than one variable.

I will describe Lowner's results, and our extensions of them.