A space-filling curve is a continuous, surjective map from [0,1] to a d-dimensional unit volume (for example, a cube or a simplex). Space-filling curves are usually constructed following a recursive tessellation of the unit volume that gives the curve useful structural properties. The most prominent of these properties is that the curve tends to preserve locality: points that are close to each other along the curve are (usually) close to each other in d-dimensional space and (usually) vice versa. This can be exploited to speed up algorithms, in practice and sometimes even in theory, by processing or storing data points in order along the curve. In this lecture I will show how space-filling curves can be described, how they get their useful properties, and I will show examples of their applications. This brings us to the question what would be the optimal space-filling curves for these applications. We will encounter a number of open questions on tessellations in 2D and 3D and on how to measure the quality of a space-filling curve.