An embedding of metric spaces is bi-Lipschitz if it distorts distances by a fixed multiplicative factor. Which metric spaces admit a bi-Lipschitz embedding into a finite-dimensional Euclidean space? Quantitative versions of this problem arise in connection with algorithmic computer science. Bi-Lipschitz embeddability is closely connected to the differentiability real-valued Lipschitz functions. I will discuss a celebrated theorem of Cheeger (1999) on the differentiability of Lipschitz functions on metric measure spaces supporting a Poincare inequality (PI spaces), and its implications for bi-Lipschitz nonembeddability. I will also describe recent examples of non-Riemannian PI spaces which admit bi-Lipschitz embeddings into Euclidean spaces.