The local theory of normed spaces aims to understand the geometry of norms through their finite dimensional subspaces. As any two norms on R^n are equivalent, such questions are trivial unless one asks for quantitative bounds, at which point rich and subtle structures emerge whose study involves a wealth of deep insights and tools. In analogy to the linear theory, the local theory of metric spaces aims to understand the geometry of metrics through the quantitative behavior of their finite subsets. We cover examples of such investigations, with emphasis on how the linear and metric theories often mimic each other, yet sometimes diverge.