Abstract: The mathematical idealizations of soap films — which have the least possible area for their given boundary — are called Minimal Surfaces, and have been studied by mathematicians and physicists since the time of Euler. I will survey some of the important developments in the history of minimal surfaces, with emphasis on the recent solution (joint work with M. Karpukhin, R. Kusner, and D. Stern) on the topological realization problem for free boundary minimal surfaces in the unit 3-ball.