Abstract: An embedding of one metric space $X$ into another metric space $Y$ is a one-to-one continuous map with continuous inverse. We fix the target space $Y$, which may be a finite-dimensional Euclidean space or some other normed vector space. Natural classes of embeddings include isometric embeddings (distances are exactly preserved) and bi-Lipschitz embeddings (distances are distorted by at most a fixed multiplicative constant). In this talk, we'll discuss the existence of embeddings within a fixed class into a fixed target space. For instance, which metric spaces admit an isometric embedding into a finite-dimensional Euclidean space? To indicate the subtlety of the problem, note that every metric space with at most three points embeds isometrically in the plane, but there exist four-point metric spaces which do not embed isometrically into any Hilbert space. Rademacher's theorem on the almost everywhere differentiability of Lipschitz functions plays a starring role. We'll highlight the use of Rademacher-type differentiation theorems for Lipschitz mappings of metric measure spaces in the proof of bi-Lipschitz nonembeddability theorems. We'll also discuss several recent bi-Lipschitz embedding theorems proved by current and former graduate students in this department.