Abstract: In this talk we will discuss the problem of finding geometric invariants of metric spaces which can be used to prove bi-Lipschitz and coarse non-embeddability results. This topic is largely motivated by recent applications to geometric group theory and theoretical computer science, as well as Banach space theory, from which many of the problems and methods originated. In particular, we will discuss ways to transfer the theory of type, cotype and uniform convexity to the context of arbitrary metric spaces, and present some applications to embedding theory and the Lipschitz extension problem.