The process by which an impurity is dispersed in a stirred fluid is important for many areas of science and engineering. It can occur as a consequence of turbulence (e.g. in stirring coffee), but mixing also occurs even in some laminar flows that cause nearby fluid elements to separate exponentially in time. This process is sometimes called "chaotic mixing". Until recently, the geometrical concepts that are important in understanding chaotic processes have not been measurable experimentally. In this talk, I show that delicate particle tracking methods now make it possible to measure the exponential separation rate of nearby fluid elements at each point in a moving fluid. We show how these "stretching fields" provide new insights into fluid mixing, and an ability to predict the rate at which stirring will cause a mixture to become homogeneous. In recent work, we extend these studies to chemically reacting mixtures, and thereby study the patterns produced as a chemical reaction proceeds. Finally, I explain why it is so important that fluid and nonlinear phenomena be included in our undergraduate science programs.