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.