In this talk I will discuss some results with Brett Wick and Ryan Hamilton that relate Nevanlinna-Pick interpolation and Toeplitz corona problems. The first of these is an interpolation problem that asks for conditions for the existence of a analytic function of norm 1 that interpolates given points in the disk to another set of given points in the disk. The second of these problems relates to the operator version of the classical Corona problem of Carleson which states that the unit disk is dense in the maximal ideal space of the algebra of bounded analytic functions on the disk. The talk will be mostly introductory and accessible to graduate students in analysis.