In this lecture, John Milnor, Co-Director of the Institute for Mathematical Sciences at Stony Brook University and a former member of the Faculty of the School of Mathematics at the Institute for Advanced Study (1970–90), offers commentary on the book On Growth and Form by D'Arcy Thompson, first published in 1917. It has remained relevant in part because it is full of examples and questions about the nature of shape and how it changes during growth and evolution. In particular, Milnor examines Thompson's suggestion that the shapes of closely related species can be transformed into one another via conformal transformation. PDF slides may be found at http://www.math.sunysb.edu/~jack/ The lecture was part of the Institute for Advanced Study's celebration of its eightieth anniversary.