Let f be a smooth function on the real line. The divided difference matrices of order n, whose, (i,j)-th entries are the divided 
 differences of f at (\lambda_i,\lambda_j) -- where \lambda_1,...,\lambda_n are prescribed real numbers -- are called Loewner 
 matrices. In a seminal paper published in 1934 Loewner used properties of these matrices to characterise operator monotone 
 functions. In the same paper he established connections between this matrix problem, complex analytic functions, and harmonic 
 analysis. These elegant connections sent Loewner matrices into the background. Some recent work has brought them back into 
 focus. In particular, characterisation of operator convex functions in terms of Loewner matrices has been obtained. In this 
 talk we describe some of this work.