The usual objects of study in classical algebraic geometry are smooth algebraic varieties. Singularities are a nuisance, to be gotten rid of by resolution or deformation. But, somewhat surprisingly, it seems that non-reduced curves (i.e. curves that are singular *everywhere*) can be an effective tool to resolve some notoriously hard questions about smooth curves. In this talk, I will introduce the simplest kinds of non-reduced curves, and explain what they tell us about the geometry of smooth curves.