A metric space is a set together with a notion of distance. An example would be 3-space with our usual definition of distance, but there are lots of examples which could be quite abstract. Suppose we're given two such spaces: how far apart are they? Does this even make sense? Is there a well-defined notion of the distance between abstract metric spaces? Can a sequence of abstract metric spaces converge? We will discuss these questions in relation to some recent research on curvature flows and geometry.