We will discuss recent developments in the theory of first-order calculus for Lipschitz functions on metric spaces that are non-smooth, abstract, or otherwise highly non-Euclidean, a theory that begins primarily with a 1999 work of Cheeger. We will explain aspects of the relationship between the geometry and topology of a metric space and the analytic structure it supports. Applications, relationships, and examples will appear from metric embedding theory, hyperbolic geometry, sub-Riemannian manifolds, and quasiconformal geometry.