In the early 20th century, Nevanlinna, L\"{o}wner, and Kraus established useful integral representations that characterize matrix monotone and matrix convex functions in one variable. Structured representations of these types have a rich history in operator theoretic approaches to complex function theory in several variables and have played a key role in the development of noncommutative function theory over the last decade. We will discuss earlier results and some recent advances based on a complex analytic approach to constructing these realizations, with attention to the connections to the classical theory.