The topology of smooth manifolds is governed largely by geometry in low dimensions and by algebraic topology in high dimensions. The phase transition occurs in dimension four, leading to “exotic” phenomena where continuous and differential topology diverge sharply. I will begin by surveying some ways that surfaces can be used to investigate this phase transition. Then I will explain how braids can shed light on surfaces in 4-manifolds, especially complex curves. In particular, I will describe my recent work revealing the emergence of exotic phenomena even in the rigid setting of complex curves.