What is the surface of the smallest area spanning a given contour? The answer can be found by dipping a wire contour into soapy water and observing the resulting soap film, a physical example of minimal surface. The technical definition of a minimal surface — that at each point the mean curvature is zero — results in the area minimizing property for any contour cut from the surface. Minimizing area goes hand in hand with minimizing energy of a structure, and consequently minimal surfaces frequently appear in nature. Minimal surfaces find application in areas as diverse as architectural tensile structures (the Munich Olympic Stadium or for a local example, the Denver International Airport) and molecular engineering, and have received much attention in recent decades. In 1982, Costa discovered the first new minimal surface in over 100 years, with extraordinary complex topology. Jane’s research as a complex analyst has led her, using essentially classical techniques, to construct new minimal surfaces with “non-convex shadows” that generalize the classical Scherk surfaces. Due to their broad application and the current level of research activity, minimal surfaces form an exciting area of modern research, both pure and applied, in the new millennium.