I will give an introduction to quantitative K-theory, K-homology and their applications. In particular, I will discuss my recent joint work with Rufus Willett on the universal coefficient theorem for nuclear C*-algebras. If time allows, I will also talk about other recent applications concerning positive scalar curvature (joint work with Hao Guo and Zhizhang Xie) and the stability of discrete groups (joint work with Shmuel Weinberger and Jianchao Wu). The talk will be accessible to a general audience.