Are there equations in mathematics that are just too hard to solve? Perhaps-and many of them seem to be nonlinear partial differential equations. But we cannot give up! These equations appear in nearly every scientific discipline, including physics, biology, chemistry, medicine, so we must work towards an understanding of these equations if we are going to make progress in science. Fortunately, there is a certain tool that can unwrap a large amount of the complexity of these equations, which is both elegant and powerful; namely, Fourier spectral analysis. We will see how this tool can be used to understand the behavior of equations governing heat flow, traffic flow, water waves, and flame fronts. This talk should be accessible to students who have taken calculus, but I will aim at also making it interesting to experts.