Lecture I: The invariant form of the Schwarz lemma can be interpreted to say that every holomorphic function from the disk D to itself is distance reducing in the pseudohyperbolic metric, an extremely useful property. Caratheodory (1927) showed how to port the pseudohyperbolic metric to any domain U, in one or several variables, by considering all holomorphic maps from U to D. Kobayashi later (1967) dualized the construction, by considering maps from D to U. Every holomorphic map from U_1 to U_2 is distance reducing with respect to both the the Caratheodory distance and the Kobayashi distance. In 1981, Laszlo Lempert proved the wonderful theorem that on convex domains, both these distances are equal to each other. Lecture II: We will discuss Agler's 1990 operator theory proof of Lempert's theorem, which involves a very careful analysis of two point interpolation problems, and the fact that certain two dimensional representations are contractive if and only if they are completely contractive. Lecture III: If V is a lower dimensional subset of a domain U, it has both its intrinsic Caratheodory and Kobayashi distances, and the ones it inherits from U.When are these the same? This is a complex analogue to asking when a submanifold, or subvariety, of a larger manifold is totally geodesic. The question lies somewhere on the nexus of Pick interpolation problems, von Neumann inequalities, contractive distances, and complex geometry. We will discuss what is known, including some recent results with L. Kosinski in the case that U is the polydisk.