I use interfacial flows as an introduction to self-similar phenomena and scaling, my main examples being optics (wave fronts), and thin film dynamics. I describe how similarity solutions can be used to describe singular behavior in higher dimensions, where in general different spatial directions are characterized by different scaling behavior. I then show how singular solutions develop complexity through a sequence of instabilities. Combining chaotic dynamics with a higher-dimensional description, one obtains a mechanism for spatial complexity, as it is characteristic for turbulent flows.