Many mathematical objects come in continuous families, prompting the desire to define a ``universal family'' that contains each such object exactly once up to isomorphism. When this isn't possible (because the family would be too bad to be worthwhile -- I'll talk about this behavior), we can try to come close, by including only ``stable'' objects. Frequently the universal family is constructed by starting with a bigger family that includes each object many times, then dividing by a group action that implements the isomorphisms. There are two ways to do this, one algebro-geometric (complex) and one symplecto-geometric (real), and I'll give some idea of why they agree. The main example will be the space of $N$ ordered points on the Riemann sphere, modulo M\"obius transformations. These are unstable if