Abstract: In 1934, Loewner proved a remarkable and deep theorem about matrix monotone functions. Recently, the young Finnish mathematician, Otte Heinävarra settled a 10 year old conjecture and found a 2 page proof of a theorem in Loewner theory whose only prior proof was 35 pages. I will describe his proof and use that as an excuse to discuss matrix monotone and matrix convex functions including, if time allows, my own recent proof of Loewner’s original theorem.