Holomorphic Dynamics (in a narrow sense) is the theory of the iteration of rational maps on the Riemann sphere. It was founded in the classical work by Fatou and Julia around 1918. After about 60 years of stagnation, it was revived in the 1980s, bringing together deep ideas from Conformal and Hyperbolic Geometry, Teichmüller Theory, the Theory of Kleinian Groups, Hyperbolic Dynamics and Ergodic Theory, and Renormalization Theory from physics, illustrated with beautiful computer-generated pictures of fractal sets (such as various Julia sets and the Mandelbrot set). We will highlight some landmarks of this story.