Free probability is a non-commutative version of (classic) probability that has applications to random matrices. Interestingly enough, free probability can be used to study the asymptotic root distribution of certain basic operations with (deterministic) polynomials. The bridge between these two topics is known as finite free probability, a topic started less than a decade ago when Marcus, Spielman and Srivastava noticed that the expected characteristic polynomial of randomly rotated matrices can be understood using some classical operations on polynomials, studied a century ago by Szegö and Walsh. The main goal of this talk is to survey the areas of free probability and finite free probability, including the connection between: -repeated differentiation and free fractional convolution. -ratio of consecutive coefficients and Voiculescu's S-transform. Time permitting we will discuss an ongoing project on the study of repeated polar differentiation.