Given a property P of graphs, it is natural to wonder how likely a given finite graph satisfies property P, that is, given some fixed natural number n, what proportion of the n vertex graphs satisfies property P.  If P is a "first-order" property, then there is a 0-1 law that says the proportion of graphs of size n that satisfy P approaches either 0 or 1 as n tends to infinity.  A simple proof of this fact uses the model theory of the so-called random (or Rado) graph.  We will present the proof of this result and then discuss recent work, joint with Bradd Hart and Alex Kruckman, which show these ideas can be used to prove an approximate 0-1 law for finite metric spaces.  The metric space version of the random graph that is relevant turns out to be somewhat surprising!